Number Fields¶

AUTHORS:

William Stein (2004, 2005): initial version
Steven Sivek (2006-05-12): added support for relative extensions
William Stein (2007-09-04): major rewrite and documentation
Robert Bradshaw (2008-10): specified embeddings into ambient fields

Note

Unlike in PARI/GP, class group computations in Sage do not by default assume the Generalized Riemann Hypothesis. To do class groups computations not provably correctly you must often pass the flag proof=False to functions or call the function proof.number_field(False). It can easily take 1000’s of times longer to do computations with proof=True (the default).

This example follows one in the Magma reference manual:

sage: K.<y> = NumberField(x^4 - 420*x^2 + 40000)
sage: z = y^5/11; z
420/11*y^3 - 40000/11*y
sage: R.<y> = PolynomialRing(K)
sage: f = y^2 + y + 1
sage: L.<a> = K.extension(f); L
Number Field in a with defining polynomial y^2 + y + 1 over its base field
sage: KL.<b> = NumberField([x^4 - 420*x^2 + 40000, x^2 + x + 1]); KL
Number Field in b0 with defining polynomial x^4 - 420*x^2 + 40000 over its base field

We do some arithmetic in a tower of relative number fields:

sage: K.<cuberoot2> = NumberField(x^3 - 2)
sage: L.<cuberoot3> = K.extension(x^3 - 3)
sage: S.<sqrt2> = L.extension(x^2 - 2)
sage: S
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
sage: sqrt2 * cuberoot3
cuberoot3*sqrt2
sage: (sqrt2 + cuberoot3)^5
(20*cuberoot3^2 + 15*cuberoot3 + 4)*sqrt2 + 3*cuberoot3^2 + 20*cuberoot3 + 60
sage: cuberoot2 + cuberoot3
cuberoot3 + cuberoot2
sage: cuberoot2 + cuberoot3 + sqrt2
sqrt2 + cuberoot3 + cuberoot2
sage: (cuberoot2 + cuberoot3 + sqrt2)^2
(2*cuberoot3 + 2*cuberoot2)*sqrt2 + cuberoot3^2 + 2*cuberoot2*cuberoot3 + cuberoot2^2 + 2
sage: cuberoot2 + sqrt2
sqrt2 + cuberoot2
sage: a = S(cuberoot2); a
cuberoot2
sage: a.parent()
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field

Warning

Doing arithmetic in towers of relative fields that depends on canonical coercions is currently VERY SLOW. It is much better to explicitly coerce all elements into a common field, then do arithmetic with them there (which is quite fast).

TESTS:

sage: y = polygen(QQ,'y'); K.<beta> = NumberField([y^3 - 3, y^2 - 2])
sage: K(y^10)
27*beta0
sage: beta^10
27*beta0

sage.rings.number_field.number_field.CyclotomicField(n, names=None, embedding=True)¶

Return the n-th cyclotomic field, where n is a positive integer.

INPUT:

n - a positive integer
names - name of generator (optional - defaults to zetan).
embedding - bool or n-th root of unity in an ambient field (default True)

EXAMPLES: We create the $7$ th cyclotomic field $\QQ(\zeta_7)$ with the default generator name.

sage: k = CyclotomicField(7); k
Cyclotomic Field of order 7 and degree 6
sage: k.gen()
zeta7

The default embedding sends the generator to the complex primitive $n^{th}$ root of unity of least argument.

sage: CC(k.gen())
0.623489801858734 + 0.781831482468030*I

Cyclotomic fields are of a special type.

sage: type(k)
<class 'sage.rings.number_field.number_field.NumberField_cyclotomic_with_category'>

We can specify a different generator name as follows.

sage: k.<z7> = CyclotomicField(7); k
Cyclotomic Field of order 7 and degree 6
sage: k.gen()
z7

The $n$ must be an integer.

sage: CyclotomicField(3/2)
...
TypeError: no conversion of this rational to integer

The degree must be positive.

sage: CyclotomicField(0)
...
ValueError: n (=0) must be a positive integer

The special case $n=1$ does not return the rational numbers:

sage: CyclotomicField(1)
Cyclotomic Field of order 1 and degree 1

Due to their default embedding into $\CC$ , cyclotomic number fields are all compatible.

sage: cf30 = CyclotomicField(30)
sage: cf5 = CyclotomicField(5)
sage: cf3 = CyclotomicField(3)
sage: cf30.gen() + cf5.gen() + cf3.gen()
zeta30^6 + zeta30^5 + zeta30 - 1
sage: cf6 = CyclotomicField(6) ; z6 = cf6.0
sage: cf3 = CyclotomicField(3) ; z3 = cf3.0
sage: cf3(z6)
zeta3 + 1
sage: cf6(z3)
zeta6 - 1
sage: cf9 = CyclotomicField(9) ; z9 = cf9.0
sage: cf18 = CyclotomicField(18) ; z18 = cf18.0
sage: cf18(z9)
zeta18^2
sage: cf9(z18)
-zeta9^5
sage: cf18(z3)
zeta18^3 - 1
sage: cf18(z6)
zeta18^3
sage: cf18(z6)**2
zeta18^3 - 1
sage: cf9(z3)
zeta9^3

sage.rings.number_field.number_field.NumberField(polynomial, name=None, check=True, names=None, cache=True, embedding=None, latex_name=None)¶

Return the number field defined by the given irreducible polynomial and with variable with the given name. If check is True (the default), also verify that the defining polynomial is irreducible and over $\QQ$ .

INPUT:

polynomial - a polynomial over $\QQ$ or a number field, or a list of polynomials.
name - a string (default: ‘a’), the name of the generator
check - bool (default: True); do type checking and irreducibility checking.
embedding - image of the generator in an ambient field (default: None)

EXAMPLES:

sage: z = QQ['z'].0
sage: K = NumberField(z^2 - 2,'s'); K
Number Field in s with defining polynomial z^2 - 2
sage: s = K.0; s
s
sage: s*s    
2
sage: s^2
2

Constructing a relative number field:

sage: K.<a> = NumberField(x^2 - 2)
sage: R.<t> = K[]
sage: L.<b> = K.extension(t^3+t+a); L
Number Field in b with defining polynomial t^3 + t + a over its base field
sage: L.absolute_field('c')
Number Field in c with defining polynomial x^6 + 2*x^4 + x^2 - 2
sage: a*b
a*b
sage: L(a)
a
sage: L.lift_to_base(b^3 + b)
-a

Constructing another number field:

sage: k.<i> = NumberField(x^2 + 1)
sage: R.<z> = k[]
sage: m.<j> = NumberField(z^3 + i*z + 3)
sage: m
Number Field in j with defining polynomial z^3 + i*z + 3 over its base field

Number fields are globally unique:

sage: K.<a>= NumberField(x^3-5)     
sage: a^3
5
sage: L.<a>= NumberField(x^3-5)
sage: K is L
True

Having different defining polynomials makes the fields different:

sage: x = polygen(QQ, 'x'); y = polygen(QQ, 'y')
sage: k.<a> = NumberField(x^2 + 3)
sage: m.<a> = NumberField(y^2 + 3)
sage: k
Number Field in a with defining polynomial x^2 + 3
sage: m
Number Field in a with defining polynomial y^2 + 3

One can also define number fields with specified embeddings, may be used for arithmetic and deduce relations with other number fields which would not be valid for an abstract number field:

sage: K.<a> = NumberField(x^3-2, embedding=1.2)
sage: RR.coerce_map_from(K)
Composite map:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Real Field with 53 bits of precision
  Defn:   Generic morphism:
          From: Number Field in a with defining polynomial x^3 - 2
          To:   Real Lazy Field
          Defn: a -> 1.259921049894873?
        then
          Conversion via _mpfr_ method map:
          From: Real Lazy Field
          To:   Real Field with 53 bits of precision
sage: RR(a)
1.25992104989487
sage: 1.1 + a
2.35992104989487
sage: b = 1/(a+1); b
1/3*a^2 - 1/3*a + 1/3
sage: RR(b)
0.442493334024442
sage: L.<b> = NumberField(x^6-2, embedding=1.1)
sage: L(a)
b^2
sage: a + b
b^2 + b

Note that the image only needs to be specified to enough precision to distinguish roots, and is exactly computed to any needed precision:

sage: RealField(200)(a)
1.2599210498948731647672106072782283505702514647015079800820

One can embed into any other field:

sage: K.<a> = NumberField(x^3-2, embedding=CC.gen()-0.6)
sage: CC(a)
-0.629960524947436 + 1.09112363597172*I
sage: L = Qp(5)
sage: f = polygen(L)^3 - 2
sage: K.<a> = NumberField(x^3-2, embedding=f.roots()[0][0])
sage: a + L(1)
4 + 2*5^2 + 2*5^3 + 3*5^4 + 5^5 + 4*5^6 + 2*5^8 + 3*5^9 + 4*5^12 + 4*5^14 + 4*5^15 + 3*5^16 + 5^17 + 5^18 + 2*5^19 + O(5^20)
sage: L.<b> = NumberField(x^6-x^2+1/10, embedding=1)
sage: K.<a> = NumberField(x^3-x+1/10, embedding=b^2)
sage: a+b
b^2 + b
sage: CC(a) == CC(b)^2
True
sage: K.coerce_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^3 - x + 1/10
  To:   Number Field in b with defining polynomial x^6 - x^2 + 1/10
  Defn: a -> b^2

The QuadraticField and CyclotomicField constructors create an embedding by default unless otherwise specified.

sage: K.<zeta> = CyclotomicField(15)
sage: CC(zeta)
0.913545457642601 + 0.406736643075800*I
sage: L.<sqrtn3> = QuadraticField(-3)
sage: K(sqrtn3)
2*zeta^5 + 1
sage: sqrtn3 + zeta
2*zeta^5 + zeta + 1

An example involving a variable name that defines a function in PARI:

sage: theta = polygen(QQ, 'theta')
sage: M.<z> = NumberField([theta^3 + 4, theta^2 + 3]); M
Number Field in z0 with defining polynomial theta^3 + 4 over its base field        

TESTS:

sage: x = QQ['x'].gen()
sage: y = ZZ['y'].gen()
sage: K = NumberField(x^3 + x + 3, 'a'); K
Number Field in a with defining polynomial x^3 + x + 3
sage: K.defining_polynomial().parent()
Univariate Polynomial Ring in x over Rational Field

sage: L = NumberField(y^3 + y + 3, 'a'); L
Number Field in a with defining polynomial y^3 + y + 3
sage: L.defining_polynomial().parent()
Univariate Polynomial Ring in y over Rational Field

sage: sage.rings.number_field.number_field._nf_cache = {}
sage: K.<x> = CyclotomicField(5)[]
sage: W.<a> = NumberField(x^2 + 1); W
Number Field in a with defining polynomial x^2 + 1 over its base field
sage: sage.rings.number_field.number_field._nf_cache = {}
sage: W1 = NumberField(x^2+1,'a')
sage: K.<x> = CyclotomicField(5)[]
sage: W.<a> = NumberField(x^2 + 1); W
Number Field in a with defining polynomial x^2 + 1 over its base field

sage.rings.number_field.number_field.NumberFieldTower(v, names, check=True, embeddings=None)¶

Return the tower of number fields defined by the polynomials or number fields in the list v.

This is the field $K_0$ generated by a root of v[0] over its base field $K_1$ , which is in turn generated by a root of v[1] over its base field $K_2$ , etc, until $K_d = \QQ$ where $d$ is the size of the list $v$ . If all is False, then each v[i] must be irreducible over the previous fields. Otherwise a list of all possible fields defined by all those polynomials is output.

If names defines a variable name a, say, then the generators of the intermediate number fields are a0, a1, a2, ...

INPUT:

v - a list of polynomials or number fields
names - variable names
check - bool (default: True) only relevant if all is False. Then check irreducibility of each input polynomial.

OUTPUT: a single number field or a list of number fields

EXAMPLES:

sage: k.<a,b,c> = NumberField([x^2 + 1, x^2 + 3, x^2 + 5]); k
Number Field in a with defining polynomial x^2 + 1 over its base field
sage: a^2
-1
sage: b^2
-3
sage: c^2
-5
sage: (a+b+c)^2
(2*b + 2*c)*a + 2*c*b - 9

The Galois group is a product of 3 groups of order 2:

sage: k.galois_group(type="pari")
Galois group PARI group [8, 1, 3, "E(8)=2[x]2[x]2"] of degree 8 of the Number Field in a with defining polynomial x^2 + 1 over its base field

Repeatedly calling base_field allows us to descend the internally constructed tower of fields:

sage: k.base_field()
Number Field in b with defining polynomial x^2 + 3 over its base field
sage: k.base_field().base_field()
Number Field in c with defining polynomial x^2 + 5
sage: k.base_field().base_field().base_field()
Rational Field

In the following example the second polynomial is reducible over the first, so we get an error:

sage: v = NumberField([x^3 - 2, x^3 - 2], names='a')
...
ValueError: defining polynomial (x^3 - 2) must be irreducible

We mix polynomial parent rings:

sage: k.<y> = QQ[]
sage: m = NumberField([y^3 - 3, x^2 + x + 1, y^3 + 2], 'beta')
sage: m
Number Field in beta0 with defining polynomial y^3 - 3 over its base field
sage: m.base_field ()
Number Field in beta1 with defining polynomial x^2 + x + 1 over its base field

A tower of quadratic fields:

sage: K.<a> = NumberField([x^2 + 3, x^2 + 2, x^2 + 1])
sage: K
Number Field in a0 with defining polynomial x^2 + 3 over its base field
sage: K.base_field()
Number Field in a1 with defining polynomial x^2 + 2 over its base field
sage: K.base_field().base_field()
Number Field in a2 with defining polynomial x^2 + 1

A bigger tower of quadratic fields.

sage: K.<a2,a3,a5,a7> = NumberField([x^2 + p for p in [2,3,5,7]]); K
Number Field in a2 with defining polynomial x^2 + 2 over its base field
sage: a2^2
-2
sage: a3^2
-3
sage: (a2+a3+a5+a7)^3
((6*a5 + 6*a7)*a3 + 6*a7*a5 - 47)*a2 + (6*a7*a5 - 45)*a3 - 41*a5 - 37*a7

class sage.rings.number_field.number_field.NumberField_absolute(polynomial, name, latex_name=None, check=True, embedding=None)¶

Bases: sage.rings.number_field.number_field.NumberField_generic

Minkowski_embedding(B=None, prec=None)¶

Return an nxn matrix over RDF whose columns are the images of the basis $\{1, \alpha, \dots, \alpha^{n-1}\}$ of self over $\QQ$ (as vector spaces), where here $\alpha$ is the generator of self over $\QQ$ , i.e. self.gen(0). If B is not None, return the images of the vectors in B as the columns instead. If prec is not None, use RealField(prec) instead of RDF.

This embedding is the so-called “Minkowski embedding” of a number field in $\RR^n$ : given the $n$ embeddings $\sigma_1, \dots, \sigma_n$ of self in $\CC$ , write $\sigma_1, \dots, \sigma_r$ for the real embeddings, and $\sigma_{r+1}, \dots, \sigma_{r+s}$ for choices of one of each pair of complex conjugate embeddings (in our case, we simply choose the one where the image of $\alpha$ has positive real part). Here $(r,s)$ is the signature of self. Then the Minkowski embedding is given by

$x \mapsto ( \sigma_1(x), \dots, \sigma_r(x), \sqrt{2}\Re(\sigma_{r+1}(x)), \sqrt{2}\Im(\sigma_{r+1}(x)), \dots, \sqrt{2}\Re(\sigma_{r+s}(x)), \sqrt{2}\Im(\sigma_{r+s}(x)))$

Equivalently, this is an embedding of self in $\RR^n$ so that the usual norm on $\RR^n$ coincides with $|x| = \sum_i |\sigma_i(x)|^2$ on self.

TODO: This could be much improved by implementing homomorphisms over VectorSpaces.

EXAMPLES:

sage: F.<alpha> = NumberField(x^3+2)
sage: F.Minkowski_embedding()
[ 1.00000000000000 -1.25992104989487  1.58740105196820]
[ 1.41421356237... 0.8908987181... -1.12246204830...]
[0.000000000000000  1.54308184421...  1.94416129723...]
sage: F.Minkowski_embedding([1, alpha+2, alpha^2-alpha])
[ 1.00000000000000 0.740078950105127  2.84732210186307]
[ 1.41421356237...  3.7193258428... -2.01336076644...]
[0.000000000000000  1.54308184421... 0.40107945302...]
sage: F.Minkowski_embedding() * (alpha + 2).vector().transpose()
[0.740078950105127]
[ 3.7193258428...]
[ 1.54308184421...]

absolute_degree()¶

A synonym for degree.

EXAMPLES:

sage: K.<i> = NumberField(x^2 + 1)
sage: K.absolute_degree()
2

absolute_different()¶

A synonym for different.

EXAMPLES:

sage: K.<i> = NumberField(x^2 + 1)
sage: K.absolute_different()
Fractional ideal (2)

absolute_discriminant()¶

A synonym for discriminant.

EXAMPLES:

sage: K.<i> = NumberField(x^2 + 1)
sage: K.absolute_discriminant()
-4

absolute_generator()¶

An alias for gen(). This is provided for consistency with relative fields, where the element returned by gen() only generates the field over its base field (not necessarily over $\QQ$ ).

EXAMPLE:

sage: K.<a> = NumberField(x^2 - 17)
sage: K.absolute_generator()
a

absolute_polynomial()¶

A synonym for polynomial.

EXAMPLES:

sage: K.<i> = NumberField(x^2 + 1)
sage: K.absolute_polynomial()
x^2 + 1

absolute_vector_space()¶

A synonym for vector_space.

EXAMPLES:

sage: K.<i> = NumberField(x^2 + 1)
sage: K.absolute_vector_space()
(Vector space of dimension 2 over Rational Field,
 Isomorphism map:
  From: Vector space of dimension 2 over Rational Field
  To:   Number Field in i with defining polynomial x^2 + 1,
 Isomorphism map:
  From: Number Field in i with defining polynomial x^2 + 1
  To:   Vector space of dimension 2 over Rational Field)

automorphisms()¶

Compute all Galois automorphisms of self.

This uses PARI’s nfgaloisconj and is much faster than root finding for many fields.

EXAMPLES:

sage: K.<a> = NumberField(x^2 + 10000)
sage: K.automorphisms()
[
Ring endomorphism of Number Field in a with defining polynomial x^2 + 10000
  Defn: a |--> a,
Ring endomorphism of Number Field in a with defining polynomial x^2 + 10000
  Defn: a |--> -a
]

Here’s a larger example, that would take some time if we found roots instead of using PARI’s specialized machinery:

sage: K = NumberField(x^6 - x^4 - 2*x^2 + 1, 'a')
sage: len(K.automorphisms())
2

$L$ is the Galois closure of $K$ :

sage: L = NumberField(x^24 - 84*x^22 + 2814*x^20 - 15880*x^18 - 409563*x^16 - 8543892*x^14 + 25518202*x^12 + 32831026956*x^10 - 672691027218*x^8 - 4985379093428*x^6 + 320854419319140*x^4 + 817662865724712*x^2 + 513191437605441, 'a')
sage: len(L.automorphisms())
24

base_field()¶

Returns the base field of self, which is always QQ

EXAMPLES:

sage: K = CyclotomicField(5)
sage: K.base_field()
Rational Field

change_names(names)¶

Return number field isomorphic to self but with the given generator name.

INPUT:

names - should be exactly one variable name.

Also, K.structure() returns from_K and to_K, where from_K is an isomorphism from K to self and to_K is an isomorphism from self to K.

EXAMPLES:

sage: K.<z> = NumberField(x^2 + 3); K
Number Field in z with defining polynomial x^2 + 3
sage: L.<ww> = K.change_names()
sage: L
Number Field in ww with defining polynomial x^2 + 3
sage: L.structure()[0]
Isomorphism given by variable name change map:
  From: Number Field in ww with defining polynomial x^2 + 3
  To:   Number Field in z with defining polynomial x^2 + 3
sage: L.structure()[0](ww + 5/3)
z + 5/3

embeddings(K)¶

Compute all field embeddings of self into the field K (which need not even be a number field, e.g., it could be the complex numbers). This will return an identical result when given K as input again.

If possible, the most natural embedding of self into K is put first in the list.

INPUT:

K - a number field

EXAMPLES:

sage: K.<a> = NumberField(x^3 - 2)
sage: L.<a1> = K.galois_closure(); L
Number Field in a1 with defining polynomial x^6 + 40*x^3 + 1372
sage: K.embeddings(L)[0]
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Number Field in a1 with defining polynomial x^6 + 40*x^3 + 1372
  Defn: a |--> 1/84*a1^4 + 13/42*a1
sage: K.embeddings(L)  is K.embeddings(L)
True

We embed a quadratic field into a cyclotomic field:

sage: L.<a> = QuadraticField(-7)
sage: K = CyclotomicField(7)
sage: L.embeddings(K)
[
Ring morphism:
  From: Number Field in a with defining polynomial x^2 + 7
  To:   Cyclotomic Field of order 7 and degree 6
  Defn: a |--> 2*zeta7^4 + 2*zeta7^2 + 2*zeta7 + 1,
Ring morphism:
  From: Number Field in a with defining polynomial x^2 + 7
  To:   Cyclotomic Field of order 7 and degree 6
  Defn: a |--> -2*zeta7^4 - 2*zeta7^2 - 2*zeta7 - 1
]

We embed a cubic field in the complex numbers:

sage: K.<a> = NumberField(x^3 - 2)
sage: K.embeddings(CC)
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.62996052494743... - 1.09112363597172*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.62996052494743... + 1.09112363597172*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 1.25992104989487
]

galois_closure(names=None, map=False)¶

Return number field $K$ that is the Galois closure of self, i.e., is generated by all roots of the defining polynomial of self, and possibly an embedding of self into $K$ .

INPUT:

names - variable name for Galois closure
map - (default: False) also return an embedding of self into $K$

EXAMPLES:

sage: K.<a> = NumberField(x^4 - 2)
sage: M = K.galois_closure('b'); M
Number Field in b with defining polynomial x^8 + 28*x^4 + 2500
sage: L.<a2> = K.galois_closure(); L
Number Field in a2 with defining polynomial x^8 + 28*x^4 + 2500
sage: K.galois_group(names=("a3")).order()
8

sage: phi = K.embeddings(L)[0]
sage: phi(K.0)
1/120*a2^5 + 19/60*a2
sage: phi(K.0).minpoly()
x^4 - 2

sage: L, phi = K.galois_closure('b', map=True)
sage: L
Number Field in b with defining polynomial x^8 + 28*x^4 + 2500
sage: phi
Ring morphism:
  From: Number Field in a with defining polynomial x^4 - 2
  To:   Number Field in b with defining polynomial x^8 + 28*x^4 + 2500
  Defn: a |--> 1/240*b^5 - 41/120*b

TESTS:

Let’s make sure we’re renaming correctly:

sage: L, phi = K.galois_closure('cc', map=True)
sage: L
Number Field in cc with defining polynomial x^8 + 28*x^4 + 2500
sage: phi
Ring morphism:
  From: Number Field in a with defining polynomial x^4 - 2
  To:   Number Field in cc with defining polynomial x^8 + 28*x^4 + 2500
  Defn: a |--> 1/240*cc^5 - 41/120*cc

is_absolute()¶

Returns True since self is an absolute field.

EXAMPLES:

sage: K = CyclotomicField(5)
sage: K.is_absolute()
True

maximal_order(v=None)¶

Return the maximal order, i.e., the ring of integers, associated to this number field.

INPUT:

v - (default: None) None, a prime, or a list of primes.
- if v is None, return the maximal order.
- if v is a prime, return an order that is p-maximal.
- if v is a list, return an order that is maximal at each prime in the list v.

EXAMPLES: In this example, the maximal order cannot be generated by a single element.

sage: k.<a> = NumberField(x^3 + x^2 - 2*x+8)
sage: o = k.maximal_order()
sage: o
Maximal Order in Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8

We compute $p$ -maximal orders for several $p$ . Note that computing a $p$ -maximal order is much faster in general than computing the maximal order:

sage: p = next_prime(10^22); q = next_prime(10^23)
sage: K.<a> = NumberField(x^3 - p*q)
sage: K.maximal_order([3]).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]
sage: K.maximal_order([2]).basis()
[1, a, a^2]
sage: K.maximal_order([p]).basis()
[1, a, a^2]
sage: K.maximal_order([q]).basis()
[1, a, a^2]
sage: K.maximal_order([p,3]).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]

An example with bigger discriminant:

sage: p = next_prime(10^97); q = next_prime(10^99)
sage: K.<a> = NumberField(x^3 - p*q)
sage: K.maximal_order(prime_range(10000)).basis()
[1, a, a^2]

optimized_representation(names=None, both_maps=True)¶

Return a field isomorphic to self with a better defining polynomial if possible, along with field isomorphisms from the new field to self and from self to the new field.

EXAMPLES: We construct a compositum of 3 quadratic fields, then find an optimized representation and transform elements back and forth.

sage: K = NumberField([x^2 + p for p in [5, 3, 2]],'a').absolute_field('b'); K
Number Field in b with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576
sage: L, from_L, to_L = K.optimized_representation()
sage: L    # your answer may different, since algorithm is random
Number Field in a14 with defining polynomial x^8 + 4*x^6 + 7*x^4 + 36*x^2 + 81
sage: to_L(K.0)   # random
4/189*a14^7 - 1/63*a14^6 + 1/27*a14^5 + 2/9*a14^4 - 5/27*a14^3 + 8/9*a14^2 + 3/7*a14 + 3/7
sage: from_L(L.0)   # random
1/1152*a1^7 + 1/192*a1^6 + 23/576*a1^5 + 17/96*a1^4 + 37/72*a1^3 + 5/6*a1^2 + 55/24*a1 + 3/4

The transformation maps are mutually inverse isomorphisms.

sage: from_L(to_L(K.0))
b
sage: to_L(from_L(L.0))     # random
a14

optimized_subfields(degree=0, name=None, both_maps=True)¶

Return optimized representations of many (but not necessarily all!) subfields of self of the given degree, or of all possible degrees if degree is 0.

EXAMPLES:

sage: K = NumberField([x^2 + p for p in [5, 3, 2]],'a').absolute_field('b'); K
Number Field in b with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576
sage: L = K.optimized_subfields(name='b')
sage: L[0][0]
Number Field in b0 with defining polynomial x - 1
sage: L[1][0]
Number Field in b1 with defining polynomial x^2 - x + 1
sage: [z[0] for z in L]          # random -- since algorithm is random
[Number Field in b0 with defining polynomial x - 1,
 Number Field in b1 with defining polynomial x^2 - x + 1,
 Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25,
 Number Field in b3 with defining polynomial x^4 - 2*x^2 + 4,
 Number Field in b4 with defining polynomial x^8 + 4*x^6 + 7*x^4 + 36*x^2 + 81]

We examine one of the optimized subfields in more detail:

sage: M, from_M, to_M = L[2]
sage: M                             # random
Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25
sage: from_M     # may be slightly random
Ring morphism:
  From: Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25
  To:   Number Field in a1 with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576
  Defn: b2 |--> -5/1152*a1^7 + 1/96*a1^6 - 97/576*a1^5 + 17/48*a1^4 - 95/72*a1^3 + 17/12*a1^2 - 53/24*a1 - 1

The to_M map is None, since there is no map from K to M:

sage: to_M

We apply the from_M map to the generator of M, which gives a rather large element of $K$ :

sage: from_M(M.0)          # random
-5/1152*a1^7 + 1/96*a1^6 - 97/576*a1^5 + 17/48*a1^4 - 95/72*a1^3 + 17/12*a1^2 - 53/24*a1 - 1

Nevertheless, that large-ish element lies in a degree 4 subfield:

sage: from_M(M.0).minpoly()   # random
x^4 - 5*x^2 + 25

order(*args, **kwds)¶

Return the order with given ring generators in the maximal order of this number field.

INPUT:

gens - list of elements of self; if no generators are given, just returns the cardinality of this number field (oo) for consistency.
check_is_integral - bool (default: True), whether to check that each generator is integral.
check_rank - bool (default: True), whether to check that the ring generated by gens is of full rank.
allow_subfield - bool (default: False), if True and the generators do not generate an order, i.e., they generate a subring of smaller rank, instead of raising an error, return an order in a smaller number field.

EXAMPLES:

sage: k.<i> = NumberField(x^2 + 1)
sage: k.order(2*i)
Order in Number Field in i with defining polynomial x^2 + 1
sage: k.order(10*i)
Order in Number Field in i with defining polynomial x^2 + 1
sage: k.order(3)
...
ValueError: the rank of the span of gens is wrong
sage: k.order(i/2)
...
ValueError: each generator must be integral

Alternatively, an order can be constructed by adjoining elements to $\ZZ$ :

sage: K.<a> = NumberField(x^3 - 2) sage: ZZ[a] Order in Number Field in a0 with defining polynomial x^3 - 2

TESTS:

We verify that trac #2480 is fixed:

sage: K.<a> = NumberField(x^4 + 4*x^2 + 2)
sage: B = K.integral_basis()
sage: K.order(*B)
Order in Number Field in a with defining polynomial x^4 + 4*x^2 + 2
sage: K.order(B)
Order in Number Field in a with defining polynomial x^4 + 4*x^2 + 2
sage: K.order(gens=B)
Order in Number Field in a with defining polynomial x^4 + 4*x^2 + 2

places(all_complex=False, prec=None)¶

Return the collection of all infinite places of self.

By default, this returns the set of real places as homomorphisms into RIF first, followed by a choice of one of each pair of complex conjugate homomorphisms into CIF.

On the other hand, if prec is not None, we simply return places into RealField(prec) and ComplexField(prec) (or RDF, CDF if prec=53). One can also use prec=infinity, which returns embeddings into the field $\overline{\QQ}$ of algebraic numbers (or its subfield $\mathbb{A}$ of algebraic reals); this permits exact computatation, but can be extremely slow.

There is an optional flag all_complex, which defaults to False. If all_complex is True, then the real embeddings are returned as embeddings into CIF instead of RIF.

EXAMPLES:

sage: F.<alpha> = NumberField(x^3-100*x+1) ; F.places()
[Ring morphism:
From: Number Field in alpha with defining polynomial x^3 - 100*x + 1
To:   Real Field with 106 bits of precision
Defn: alpha |--> -10.00499625499181184573367219280,
Ring morphism:
From: Number Field in alpha with defining polynomial x^3 - 100*x + 1
To:   Real Field with 106 bits of precision
Defn: alpha |--> 0.01000001000003000012000055000273,
Ring morphism:
From: Number Field in alpha with defining polynomial x^3 - 100*x + 1
To:   Real Field with 106 bits of precision
Defn: alpha |--> 9.994996244991781845613530439509]

sage: F.<alpha> = NumberField(x^3+7) ; F.places()
[Ring morphism:
From: Number Field in alpha with defining polynomial x^3 + 7
To:   Real Field with 106 bits of precision
Defn: alpha |--> -1.912931182772389101199116839549,
Ring morphism:
From: Number Field in alpha with defining polynomial x^3 + 7
To:   Complex Field with 53 bits of precision
Defn: alpha |--> 0.956465591386195 + 1.65664699997230*I]

sage: F.<alpha> = NumberField(x^3+7) ; F.places(all_complex=True)
[Ring morphism:
From: Number Field in alpha with defining polynomial x^3 + 7
To:   Complex Field with 53 bits of precision
Defn: alpha |--> -1.91293118277239,
Ring morphism:
From: Number Field in alpha with defining polynomial x^3 + 7
To:   Complex Field with 53 bits of precision
Defn: alpha |--> 0.956465591386195 + 1.65664699997230*I]
sage: F.places(prec=10)
[Ring morphism:
From: Number Field in alpha with defining polynomial x^3 + 7
To:   Real Field with 10 bits of precision
Defn: alpha |--> -1.9,
Ring morphism:
From: Number Field in alpha with defining polynomial x^3 + 7
To:   Complex Field with 10 bits of precision
Defn: alpha |--> 0.96 + 1.7*I]

real_places(prec=None)¶

Return all real places of self as homomorphisms into RIF.

EXAMPLES:

sage: F.<alpha> = NumberField(x^4-7) ; F.real_places()
[Ring morphism:
From: Number Field in alpha with defining polynomial x^4 - 7
To:   Real Field with 106 bits of precision
Defn: alpha |--> -1.626576561697785743211232345494,
Ring morphism:
From: Number Field in alpha with defining polynomial x^4 - 7
To:   Real Field with 106 bits of precision
Defn: alpha |--> 1.626576561697785743211232345494]

relative_degree()¶

A synonym for degree.

EXAMPLES:

sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_degree()
2

relative_different()¶

A synonym for different.

EXAMPLES:

sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_different()
Fractional ideal (2)

relative_discriminant()¶

A synonym for discriminant.

EXAMPLES:

sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_discriminant()
-4

relative_polynomial()¶

A synonym for polynomial.

EXAMPLES:

sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_polynomial()
x^2 + 1

relative_vector_space()¶

A synonym for vector_space.

EXAMPLES:

sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_vector_space()
(Vector space of dimension 2 over Rational Field,
 Isomorphism map:
  From: Vector space of dimension 2 over Rational Field
  To:   Number Field in i with defining polynomial x^2 + 1,
 Isomorphism map:
  From: Number Field in i with defining polynomial x^2 + 1
  To:   Vector space of dimension 2 over Rational Field)

relativize(alpha, names)¶

Given an element in self or an embedding of a subfield into self, return a relative number field $K$ isomorphic to self that is relative over the absolute field $\QQ(\alpha)$ or the domain of $alpha$ , along with isomorphisms from $K$ to self and from self to $K$ .

INPUT:

alpha - an element of self or an embedding of a subfield into self
names - 2-tuple of names of generator for output field K and the subfield QQ(alpha) names[0] generators K and names[1] QQ(alpha).

OUTPUT:

K – relative number field

Also, K.structure() returns from_K and to_K, where from_K is an isomorphism from K to self and to_K is an isomorphism from self to K.

EXAMPLES:

sage: K.<a> = NumberField(x^10 - 2)
sage: L.<c,d> = K.relativize(a^4 + a^2 + 2); L
Number Field in c with defining polynomial x^2 - 1/5*d^4 + 8/5*d^3 - 23/5*d^2 + 7*d - 18/5 over its base field
sage: c.absolute_minpoly()
x^10 - 2
sage: d.absolute_minpoly()
x^5 - 10*x^4 + 40*x^3 - 90*x^2 + 110*x - 58
sage: (a^4 + a^2 + 2).minpoly()
x^5 - 10*x^4 + 40*x^3 - 90*x^2 + 110*x - 58
sage: from_L, to_L = L.structure()
sage: to_L(a)
c
sage: to_L(a^4 + a^2 + 2)
d
sage: from_L(to_L(a^4 + a^2 + 2))
a^4 + a^2 + 2

The following demonstrates distinct embeddings of a subfield into a larger field:

sage: K.<a> = NumberField(x^4 + 2*x^2 + 2)
sage: K0 = K.subfields(2)[0][0]; K0
Number Field in a0 with defining polynomial x^2 - 2*x + 2
sage: rho, tau = K0.embeddings(K)
sage: L0 = K.relativize(rho(K0.gen()), 'b'); L0
Number Field in b0 with defining polynomial x^2 - b1 + 2 over its base field
sage: L1 = K.relativize(rho, 'b'); L1
Number Field in b0 with defining polynomial x^2 - a0 + 2 over its base field
sage: L2 = K.relativize(tau, 'b'); L2
Number Field in b0 with defining polynomial x^2 + a0 over its base field
sage: L0.base_field() is K0
False
sage: L1.base_field() is K0
True
sage: L2.base_field() is K0
True

Here we see that with the different embeddings, the relative norms are different:

sage: a0 = K0.gen()
sage: L1_into_K, K_into_L1 = L1.structure()
sage: L2_into_K, K_into_L2 = L2.structure()
sage: len(K.factor(41))
4
sage: w1 = -a^2 + a + 1; P = K.ideal([w1])
sage: Pp = L1.ideal(K_into_L1(w1)).ideal_below(); Pp == K0.ideal([4*a0 + 1])
True
sage: Pp == w1.norm(rho)
True

sage: w2 = a^2 + a - 1; Q = K.ideal([w2])
sage: Qq = L2.ideal(K_into_L2(w2)).ideal_below(); Qq == K0.ideal([-4*a0 + 9])
True
sage: Qq == w2.norm(tau)
True

sage: Pp == Qq
False

TESTS:

We can relativize over the whole field:

sage: K.<a> = NumberField(x^4 + 2*x^2 + 2)
sage: K.relativize(K.gen(), 'a')
Number Field in a0 with defining polynomial x - a1 over its base field
sage: K.relativize(2*K.gen(), 'a')
Number Field in a0 with defining polynomial x - 1/2*a1 over its base field

We can relativize over the prime field:

sage: L = K.relativize(K(1), 'a'); L
Number Field in a0 with defining polynomial x^4 + 2*x^2 + 2 over its base field
sage: L.base_field()
Number Field in a1 with defining polynomial x - 1
sage: L.base_field().base_field()
Rational Field

sage: L = K.relativize(K(2), 'a'); L
Number Field in a0 with defining polynomial x^4 + 2*x^2 + 2 over its base field
sage: L.base_field()
Number Field in a1 with defining polynomial x - 2
sage: L.base_field().base_field()
Rational Field

We can relativize over a relative field:

sage: K.<z> = CyclotomicField(16)
sage: L, L_into_K, _ = K.subfields(4)[0]; L
Number Field in z0 with defining polynomial x^4 + 16
sage: F, F_into_L, _ = L.subfields(2)[0]; F
Number Field in z00 with defining polynomial x^2 + 64

sage: L_over_F = L.relativize(F_into_L, 'c'); L_over_F
Number Field in c0 with defining polynomial x^2 - 1/2*z00 over its base field
sage: L_over_F_into_L, _ = L_over_F.structure()

sage: K_over_rel = K.relativize(L_into_K * L_over_F_into_L, 'a'); K_over_rel
Number Field in a0 with defining polynomial x^2 - 1/2*c0 over its base field
sage: K_over_rel.base_field() is L_over_F
True
sage: K_over_rel.structure()
(Relative number field morphism:
  From: Number Field in a0 with defining polynomial x^2 - 1/2*c0 over its base field
  To:   Cyclotomic Field of order 16 and degree 8
  Defn: a0 |--> z
        c0 |--> 2*z^2
        z00 |--> 8*z^4, Ring morphism:
  From: Cyclotomic Field of order 16 and degree 8
  To:   Number Field in a0 with defining polynomial x^2 - 1/2*c0 over its base field
  Defn: z |--> a0)

We can relativize over a really large field. This requires great care to not factor or do any operation that would trigger a pari nfinit() internally:

sage: K.<a> = CyclotomicField(3^3*2^3)
sage: R = K.relativize(a^(3^2), 't'); R
Number Field in t0 with defining polynomial x^9 - t1 over its base field
sage: R.structure()
(Relative number field morphism:
  From: Number Field in t0 with defining polynomial x^9 - t1 over its base field
  To:   Cyclotomic Field of order 216 and degree 72
  Defn: t0 |--> a
        t1 |--> a^9,
 Ring morphism:
  From: Cyclotomic Field of order 216 and degree 72
  To:   Number Field in t0 with defining polynomial x^9 - t1 over its base field
  Defn: a |--> t0)

subfields(degree=0, name=None)¶

Return all subfields of self of the given degree, or of all possible degrees if degree is 0. The subfields are returned as absolute fields together with an embedding into self. For the case of the field itself, the reverse isomorphism is also provided.

EXAMPLES:

sage: K.<a> = NumberField( [x^3 - 2, x^2 + x + 1] )
sage: K = K.absolute_field('b')
sage: S = K.subfields()
sage: len(S)
6
sage: [k[0].polynomial() for k in S]
[x - 3,
 x^2 - 3*x + 9,
 x^3 - 3*x^2 + 3*x + 1,
 x^3 - 3*x^2 + 3*x + 1,
 x^3 - 3*x^2 + 3*x - 17,
 x^6 - 3*x^5 + 6*x^4 - 11*x^3 + 12*x^2 + 3*x + 1]
 sage: R.<t> = QQ[]
 sage: L = NumberField(t^3 - 3*t + 1, 'c')
 sage: [k[1] for k in L.subfields()]
 [Ring morphism:
   From: Number Field in c0 with defining polynomial t
   To:   Number Field in c with defining polynomial t^3 - 3*t + 1
   Defn: 0 |--> 0,
  Ring morphism:
   From: Number Field in c1 with defining polynomial t^3 - 3*t + 1
   To:   Number Field in c with defining polynomial t^3 - 3*t + 1
   Defn: c1 |--> c]
sage: L.subfields(2)
[]

vector_space()¶

Return a vector space V and isomorphisms self –> V and V –> self.

OUTPUT:

V - a vector space over the rational numbers
from_V - an isomorphism from V to self
to_V - an isomorphism from self to V

EXAMPLES:

sage: k.<a> = NumberField(x^3 + 2)
sage: V, from_V, to_V  = k.vector_space()
sage: from_V(V([1,2,3]))
3*a^2 + 2*a + 1
sage: to_V(1 + 2*a + 3*a^2)
(1, 2, 3)
sage: V
Vector space of dimension 3 over Rational Field
sage: to_V
Isomorphism map:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Vector space of dimension 3 over Rational Field
sage: from_V(to_V(2/3*a - 5/8))
2/3*a - 5/8
sage: to_V(from_V(V([0,-1/7,0])))
(0, -1/7, 0)

sage.rings.number_field.number_field.NumberField_absolute_v1(poly, name, latex_name, canonical_embedding=None)¶

This is used in pickling generic number fields.

EXAMPLES:

sage: from sage.rings.number_field.number_field import NumberField_generic_v1
sage: R.<x> = QQ[]
sage: NumberField_generic_v1(x^2 + 1, 'i', 'i')
Number Field in i with defining polynomial x^2 + 1

class sage.rings.number_field.number_field.NumberField_cyclotomic(n, names, embedding=None)¶

Bases: sage.rings.number_field.number_field.NumberField_absolute

Create a cyclotomic extension of the rational field.

The command CyclotomicField(n) creates the n-th cyclotomic field, obtained by adjoining an n-th root of unity to the rational field.

EXAMPLES:

sage: CyclotomicField(3)
Cyclotomic Field of order 3 and degree 2
sage: CyclotomicField(18)
Cyclotomic Field of order 18 and degree 6
sage: z = CyclotomicField(6).gen(); z
zeta6
sage: z^3
-1
sage: (1+z)^3
6*zeta6 - 3

sage: K = CyclotomicField(197)
sage: loads(K.dumps()) == K
True
sage: loads((z^2).dumps()) == z^2
True

sage: cf12 = CyclotomicField( 12 )
sage: z12 = cf12.0
sage: cf6 = CyclotomicField( 6 )
sage: z6 = cf6.0
sage: FF = Frac( cf12['x'] )
sage: x = FF.0
sage: z6*x^3/(z6 + x)
zeta12^2*x^3/(x + zeta12^2)

sage: cf6 = CyclotomicField(6) ; z6 = cf6.gen(0)
sage: cf3 = CyclotomicField(3) ; z3 = cf3.gen(0)
sage: cf3(z6)
zeta3 + 1
sage: cf6(z3)
zeta6 - 1
sage: type(cf6(z3))
<type 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic'>
sage: cf1 = CyclotomicField(1) ; z1 = cf1.0
sage: cf3(z1)
1
sage: type(cf3(z1))
<type 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic'>

complex_embedding(prec=53)¶

Return the embedding of this cyclotomic field into the approximate complex field with precision prec obtained by sending the generator $\zeta$ of self to exp(2*pi*i/n), where $n$ is the multiplicative order of $\zeta$ .

EXAMPLES:

sage: C = CyclotomicField(4)
sage: C.complex_embedding()
Ring morphism:
  From: Cyclotomic Field of order 4 and degree 2
  To:   Complex Field with 53 bits of precision
  Defn: zeta4 |--> 6.12323399573677e-17 + 1.00000000000000*I

Note in the example above that the way zeta is computed (using sin and cosine in MPFR) means that only the prec bits of the number after the decimal point are valid.

sage: K = CyclotomicField(3)
sage: phi = K.complex_embedding(10)
sage: phi(K.0)
-0.50 + 0.87*I
sage: phi(K.0^3)
1.0
sage: phi(K.0^3 - 1)
0
sage: phi(K.0^3 + 7)
8.0

complex_embeddings(prec=53)¶

Return all embeddings of this cyclotomic field into the approximate complex field with precision prec.

If you want 53-bit double precision, which is faster but less reliable, then do self.embeddings(CDF).

EXAMPLES:

sage: CyclotomicField(5).complex_embeddings()
[
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> 0.309016994374947 + 0.951056516295154*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> -0.809016994374947 + 0.587785252292473*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> -0.809016994374947 - 0.587785252292473*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> 0.309016994374947 - 0.951056516295154*I
]

different()¶

Returns the different ideal of the cyclotomic field self.

EXAMPLES:

sage: C20 = CyclotomicField(20)
sage: C20.different()
Fractional ideal (-4*zeta20^7 + 8*zeta20^5 - 12*zeta20^3 + 6*zeta20)
sage: C18 = CyclotomicField(18)
sage: D = C18.different().norm()
sage: D == C18.discriminant().abs()
True

discriminant(v=None)¶

Returns the discriminant of the ring of integers of the cyclotomic field self, or if v is specified, the determinant of the trace pairing on the elements of the list v.

Uses the formula for the discriminant of a prime power cyclotomic field and Hilbert Theorem 88 on the discriminant of composita.

INPUT:

v (optional) - list of element of this number field

OUTPUT: Integer if v is omitted, and Rational otherwise.

EXAMPLES:

sage: CyclotomicField(20).discriminant()
4000000
sage: CyclotomicField(18).discriminant()
-19683

integral_basis(v=None)¶

Return a list of elements of this number field that are a basis for the full ring of integers.

This field is cyclomotic, so this is a trivial computation, since the power basis on the generator is an integral basis. Thus the v parameter is ignored.

EXAMPLES:

sage: CyclotomicField(5).integral_basis()
[1, zeta5, zeta5^2, zeta5^3]

is_galois()¶

Return True since all cyclotomic fields are automatically Galois.

EXAMPLES:

sage: CyclotomicField(29).is_galois()
True

is_isomorphic(other)¶

Return True if the cyclotomic field self is isomorphic as a number field to other.

EXAMPLES:

sage: CyclotomicField(11).is_isomorphic(CyclotomicField(22))
True
sage: CyclotomicField(11).is_isomorphic(CyclotomicField(23))
False
sage: CyclotomicField(3).is_isomorphic(NumberField(x^2 + x +1, 'a'))
True

next_split_prime(p=2)¶

Return the next prime integer $p$ that splits completely in this cyclotomic field (and does not ramify).

EXAMPLES:

sage: K.<z> = CyclotomicField(3)
sage: K.next_split_prime(7)
13

number_of_roots_of_unity()¶

Return number of roots of unity in this cyclotomic field.

EXAMPLES:

sage: K.<a> = CyclotomicField(21)
sage: K.number_of_roots_of_unity()
42

primitive_root_of_unity()¶

Return a generator of the roots of unity in this field.

EXAMPLES:

sage: K.<a> = CyclotomicField(3)
sage: z = K.primitive_root_of_unity(); z
-a
sage: z.multiplicative_order()
6

sage: K.<a> = CyclotomicField(10)
sage: z = K.primitive_root_of_unity(); z
a
sage: z.multiplicative_order()
10

real_embeddings(prec=53)¶

Return all embeddings of this cyclotomic field into the approximate real field with precision prec.

Mostly, of course, there are no such embeddings.

EXAMPLES:

sage: CyclotomicField(4).real_embeddings()
[]
sage: CyclotomicField(2).real_embeddings()
[
Ring morphism:
  From: Cyclotomic Field of order 2 and degree 1
  To:   Real Field with 53 bits of precision
  Defn: -1 |--> -1.00000000000000
]

roots_of_unity()¶

Return all the roots of unity in this cyclotomic field, primitive or not.

EXAMPLES:

sage: K.<a> = CyclotomicField(3)
sage: zs = K.roots_of_unity(); zs
[1, a, -a - 1, -1, -a, a + 1]
sage: [ z**K.number_of_roots_of_unity() for z in zs ]
[1, 1, 1, 1, 1, 1]

signature()¶

Return (r1, r2), where r1 and r2 are the number of real embeddings and pairs of complex embeddings of this cyclotomic field, respectively.

Trivial since, apart from QQ, cyclotomic fields are totally complex.

EXAMPLES:

sage: CyclotomicField(5).signature()
(0, 2)
sage: CyclotomicField(2).signature()
(1, 0)

zeta(n=None, all=False)¶

Returns an element of multiplicative order $n$ in this this cyclotomic field, if there is one. Raises a ValueError if there is not.

INPUT:

n - integer (default: None, returns element of maximal order)
all - bool (default: False) - whether to return a list of all n-th roots.

OUTPUT: root of unity or list

EXAMPLES:

sage: k = CyclotomicField(7)
sage: k.zeta()
zeta7
sage: k.zeta().multiplicative_order()
7
sage: k = CyclotomicField(49)
sage: k.zeta().multiplicative_order()
49
sage: k.zeta(7).multiplicative_order()
7
sage: k.zeta()
zeta49
sage: k.zeta(7)
zeta49^7

sage: K.<a> = CyclotomicField(7)
sage: K.zeta(14, all=True)
[-a^4, -a^5, a^5 + a^4 + a^3 + a^2 + a + 1, -a, -a^2, -a^3]
sage: K.<a> = CyclotomicField(10)
sage: K.zeta(20, all=True)
...
ValueError: n (=20) does not divide order of generator

sage: K.<a> = CyclotomicField(5)
sage: K.zeta(4)
...
ValueError: n (=4) does not divide order of generator
sage: v = K.zeta(5, all=True); v
[a, a^2, a^3, -a^3 - a^2 - a - 1]
sage: [b^5 for b in v]
[1, 1, 1, 1]

zeta_order()¶

Return the order of the maximal root of unity contained in this cyclotomic field.

EXAMPLES:

sage: CyclotomicField(1).zeta_order()
2
sage: CyclotomicField(4).zeta_order()
4
sage: CyclotomicField(5).zeta_order()
10
sage: CyclotomicField(5)._n()
5
sage: CyclotomicField(389).zeta_order()
778

sage.rings.number_field.number_field.NumberField_cyclotomic_v1(zeta_order, name, canonical_embedding=None)¶

This is used in pickling cyclotomic fields.

EXAMPLES:

sage: from sage.rings.number_field.number_field import NumberField_cyclotomic_v1
sage: NumberField_cyclotomic_v1(5,'a')
Cyclotomic Field of order 5 and degree 4
sage: NumberField_cyclotomic_v1(5,'a').variable_name()
'a'

class sage.rings.number_field.number_field.NumberField_generic(polynomial, name, latex_name=None, check=True, embedding=None, category=None)¶

Bases: sage.rings.number_field.number_field_base.NumberField

EXAMPLES:

sage: K.<a> = NumberField(x^3 - 2); K
Number Field in a with defining polynomial x^3 - 2
sage: loads(K.dumps()) == K
True

S_class_group(S, proof=True)¶

Returns a list of representatives which generate the S-class group of this number field over its base field.

INPUT:

S - a set of primes of the base field
proof - if False, assume Pari’s GRH++ in computing the class group

OUTPUT:

A list of fractional ideal representatives of the class group generators.

EXAMPLE:

A well known example:

sage: K.<a> = QuadraticField(-5)
sage: K.S_class_group([])
[Fractional ideal (2, a + 1)]

When we include the prime $(2, a+1)$ , the S-class group becomes trivial:

sage: K.S_class_group([K.ideal(2,a+1)])
[]

S_units(S, proof=True)¶

Returns a list of generators of the S-units.

INPUT:

- ``S`` - a set of primes of the base field

proof - if False, assume Pari’s GRH++ in computing the class group

OUTPUT:

A list of generators of the unit group.

EXAMPLE:

sage: K.<a> = QuadraticField(-3)
sage: K.unit_group()
Unit group with structure C6 of Number Field in a with defining polynomial x^2 + 3
sage: K.S_units([])
[-1/2*a + 1/2]
sage: K.S_units([])[0].multiplicative_order()
6

An example in a relative extension (see trac #8722):

sage: L.<a,b> = NumberField([x^2 + 1, x^2 - 5])
sage: p = L.ideal((-1/2*b - 1/2)*a + 1/2*b - 1/2)
sage: W = L.S_units([p]); [x.norm() for x in W]
[9, 1, 1]

absolute_degree()¶

Return the degree of self over $\QQ$ .

EXAMPLES:

sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').absolute_degree()
3
sage: NumberField(x + 1, 'a').absolute_degree()
1
sage: NumberField(x^997 + 17*x + 3, 'a', check=False).absolute_degree()
997

absolute_field(names)¶

Returns self as an absolute extension over QQ.

OUTPUT:

K - this number field (since it is already absolute)

Also, K.structure() returns from_K and to_K, where from_K is an isomorphism from K to self and to_K is an isomorphism from self to K.

EXAMPLES:

sage: K = CyclotomicField(5)
sage: K.absolute_field('a')
Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1

absolute_polynomial_ntl()¶

algebraic_closure()¶

Return the algebraic closure of self (which is QQbar).

EXAMPLES:

sage: K.<i> = QuadraticField(-1)
sage: K.algebraic_closure()
Algebraic Field
sage: K.<a> = NumberField(x^3-2)
sage: K.algebraic_closure()
Algebraic Field
sage: K = CyclotomicField(23)
sage: K.algebraic_closure()
Algebraic Field

change_generator(alpha, name=None, names=None)¶

Given the number field self, construct another isomorphic number field $K$ generated by the element alpha of self, along with isomorphisms from $K$ to self and from self to $K$ .

EXAMPLES:

sage: L.<i> = NumberField(x^2 + 1); L
Number Field in i with defining polynomial x^2 + 1
sage: K, from_K, to_K = L.change_generator(i/2 + 3)
sage: K
Number Field in i0 with defining polynomial x^2 - 6*x + 37/4
sage: from_K
Ring morphism:
  From: Number Field in i0 with defining polynomial x^2 - 6*x + 37/4
  To:   Number Field in i with defining polynomial x^2 + 1
  Defn: i0 |--> 1/2*i + 3
sage: to_K
Ring morphism:
  From: Number Field in i with defining polynomial x^2 + 1
  To:   Number Field in i0 with defining polynomial x^2 - 6*x + 37/4
  Defn: i |--> 2*i0 - 6

We can also do

sage: K.<c>, from_K, to_K = L.change_generator(i/2 + 3); K
Number Field in c with defining polynomial x^2 - 6*x + 37/4

We compute the image of the generator $\sqrt{-1}$ of $L$ .

sage: to_K(i)
2*c - 6

Note that the image is indeed a square root of -1.

sage: to_K(i)^2
-1
sage: from_K(to_K(i))
i
sage: to_K(from_K(c))
c

characteristic()¶

Return the characteristic of this number field, which is of course 0.

EXAMPLES:

sage: k.<a> = NumberField(x^99 + 2); k
Number Field in a with defining polynomial x^99 + 2
sage: k.characteristic()
0

class_group(proof=None, names='c')¶

Return the class group of the ring of integers of this number field.

INPUT:

proof - if True then compute the class group provably correctly. Default is True. Call number_field_proof to change this default globally.
names - names of the generators of this class group.

OUTPUT: The class group of this number field.

EXAMPLES:

sage: K.<a> = NumberField(x^2 + 23)
sage: G = K.class_group(); G
Class group of order 3 with structure C3 of Number Field in a with defining polynomial x^2 + 23
sage: G.0
Fractional ideal class (2, 1/2*a - 1/2)
sage: G.gens()
[Fractional ideal class (2, 1/2*a - 1/2)]

sage: G.number_field()
Number Field in a with defining polynomial x^2 + 23
sage: G is K.class_group()
True
sage: G is K.class_group(proof=False)
False
sage: G.gens()
[Fractional ideal class (2, 1/2*a - 1/2)]

There can be multiple generators:

sage: k.<a> = NumberField(x^2 + 20072)
sage: G = k.class_group(); G
Class group of order 76 with structure C38 x C2 of Number Field in a with defining polynomial x^2 + 20072
sage: G.0 # random
Fractional ideal class (41, a + 10)
sage: G.0^38
Trivial principal fractional ideal class
sage: G.1 # random
Fractional ideal class (2, -1/2*a)
sage: G.1^2
Trivial principal fractional ideal class

Class groups of Hecke polynomials tend to be very small:

sage: f = ModularForms(97, 2).T(2).charpoly()
sage: f.factor()
(x - 3) * (x^3 + 4*x^2 + 3*x - 1) * (x^4 - 3*x^3 - x^2 + 6*x - 1)
sage: [NumberField(g,'a').class_group().order() for g,_ in f.factor()]
[1, 1, 1]

class_number(proof=None)¶

Return the class number of this number field, as an integer.

INPUT:

proof - bool (default: True unless you called number_field_proof)

EXAMPLES:

sage: NumberField(x^2 + 23, 'a').class_number()
3
sage: NumberField(x^2 + 163, 'a').class_number()
1
sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').class_number(proof=False)
1539

completion(p, prec, extras={})¶

Returns the completion of self at $p$ to the specified precision. Only implemented at archimedean places, and then only if an embedding has been fixed.

EXAMPLES:

sage: K.<a> = QuadraticField(2)
sage: K.completion(infinity, 100)
Real Field with 100 bits of precision
sage: K.<zeta> = CyclotomicField(12)
sage: K.completion(infinity, 53, extras={'type': 'RDF'})
Complex Double Field
sage: zeta + 1.5                            # implicit test
2.36602540378444 + 0.500000000000000*I

complex_embeddings(prec=53)¶

Return all homomorphisms of this number field into the approximate complex field with precision prec.

This always embeds into an MPFR based complex field. If you want embeddings into the 53-bit double precision, which is faster, use self.embeddings(CDF).

EXAMPLES:

sage: k.<a> = NumberField(x^5 + x + 17)
sage: v = k.complex_embeddings()
sage: ls = [phi(k.0^2) for phi in v] ; ls # random order
[2.97572074038...,
 -2.40889943716 + 1.90254105304*I,
 -2.40889943716 - 1.90254105304*I,
 0.921039066973 + 3.07553311885*I,
 0.921039066973 - 3.07553311885*I]
sage: K.<a> = NumberField(x^3 + 2)
sage: ls = K.complex_embeddings() ; ls # random order
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Complex Double Field
  Defn: a |--> -1.25992104989...,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Complex Double Field
  Defn: a |--> 0.629960524947 - 1.09112363597*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Complex Double Field
  Defn: a |--> 0.629960524947 + 1.09112363597*I
]

composite_fields(other, names=None, both_maps=False, preserve_embedding=True)¶

List of all possible composite number fields formed from self and other, together with (optionally) embeddings into the compositum; see the documentation for both_maps below.

If preserve_embedding is True and if self and other both have embeddings into the same ambient field, or into fields which are contained in a common field, only the compositum respecting both embeddings is returned. If one (or both) of self or other does not have an embedding or preserve_embedding is False, all possible composite number fields are returned.

INPUT:

other - a number field
names - generator name for composite fields
both_maps - (default: False) if True, return quadruples (F, self_into_F, other_into_F, k) such that self_into_F is an embedding of self in F, other_into_F is an embedding of in F, and k is an integer such that F.gen() equals other_into_F(other.gen()) + k*self_into_F(self.gen()) or has the value Infinity in which case F.gen() equals self_into_F(self.gen()), or is None (which happens when other is a relative number field). If both self and other have embeddings into an ambient field, then F will have an embedding with respect to which both self_into_F and other_into_F will be compatible with the ambient embeddings.
preserve_embedding - (default: True) if self and other have ambient embeddings, then return only the compatible compositum.

OUTPUT:

list - list of the composite fields, possibly with maps.

EXAMPLES:

sage: K.<a> = NumberField(x^4 - 2)
sage: K.composite_fields(K)
[Number Field in a with defining polynomial x^4 - 2, 
 Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500]

A particular compositum is selected, together with compatible maps into the compositum, if the fields are endowed with a real or complex embedding:

sage: K1 = NumberField(x^4 - 2, 'a', embedding=RR(2^(1/4)))
sage: K2 = NumberField(x^4 - 2, 'a', embedding=RR(-2^(1/4)))
sage: K1.composite_fields(K2)
[Number Field in a with defining polynomial x^4 - 2]
sage: [F, f, g, k], = K1.composite_fields(K2, both_maps=True); F
Number Field in a with defining polynomial x^4 - 2
sage: f(K1.0), g(K2.0)
(a, -a)

With preserve_embedding set to False, the embeddings are ignored:

sage: K1.composite_fields(K2, preserve_embedding=False)
[Number Field in a with defining polynomial x^4 - 2,
 Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500]

Changing the embedding selects a different compositum:

sage: K3 = NumberField(x^4 - 2, 'a', embedding=CC(2^(1/4)*I))
sage: [F, f, g, k], = K1.composite_fields(K3, both_maps=True); F
Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500
sage: f(K1.0), g(K3.0)
(1/240*a0^5 - 41/120*a0, 1/120*a0^5 + 19/60*a0)

If no embeddings are specified, the maps into the composite are chosen arbitrarily:

sage: Q1.<a> = NumberField(x^4 + 10*x^2 + 1)
sage: Q2.<b> = NumberField(x^4 + 16*x^2 + 4)
sage: Q1.composite_fields(Q2, 'c')
[Number Field in c with defining polynomial x^8 + 64*x^6 + 904*x^4 + 3840*x^2 + 3600]
sage: F, Q1_into_F, Q2_into_F, k = Q1.composite_fields(Q2, 'c', both_maps=True)[0]
sage: Q1_into_F
Ring morphism:
  From: Number Field in a with defining polynomial x^4 + 10*x^2 + 1
  To:   Number Field in c with defining polynomial x^8 + 64*x^6 + 904*x^4 + 3840*x^2 + 3600
  Defn: a |--> 19/14400*c^7 + 137/1800*c^5 + 2599/3600*c^3 + 8/15*c

This is just one of four embeddings of Q1 into F::: sage: Hom(Q1, F).order() 4

TESTS:

Let’s check that embeddings are being respected:

sage: x = ZZ['x'].0
sage: K0.<b> = CyclotomicField(7, 'a').subfields(3)[0][0].change_names()
sage: K1.<a1> = K0.extension(x^2 - 2*b^2, 'a1').absolute_field()
sage: K2.<a2> = K0.extension(x^2 - 3*b^2, 'a2').absolute_field()

We need embeddings, so we redefine:

sage: L1.<a1> = NumberField(K1.polynomial(), 'a1', embedding=CC.0)
sage: L2.<a2> = NumberField(K2.polynomial(), 'a2', embedding=CC.0)
sage: [CDF(a1), CDF(a2)]
[-0.629384245426, -0.77083512672]

and we get the same embeddings via the compositum:

sage: F, L1_into_F, L2_into_F, k = L1.composite_fields(L2, both_maps=True)[0]
sage: [CDF(L1_into_F(L1.gen())), CDF(L2_into_F(L2.gen()))]
[-0.629384245426, -0.77083512672]

Let’s check that if only one field has an embedding, the resulting fields do not have embeddings:

sage: L1.composite_fields(K2)[0].coerce_embedding() is None
True
sage: L2.composite_fields(K1)[0].coerce_embedding() is None
True

We check that other can be a relative number field:

sage: L.<a, b> = NumberField([x^3 - 5, x^2 + 3])        
sage: CyclotomicField(3, 'w').composite_fields(L, both_maps=True)
[(Number Field in a with defining polynomial x^3 - 5 over its base field, Ring morphism:
  From: Cyclotomic Field of order 3 and degree 2
  To:   Number Field in a with defining polynomial x^3 - 5 over its base field
  Defn: w |--> -1/2*b - 1/2, Relative number field endomorphism of Number Field in a with defining polynomial x^3 - 5 over its base field
  Defn: a |--> a
        b |--> b, None)]

defining_polynomial()¶

Return the defining polynomial of this number field.

This is exactly the same as self.polynomal().

EXAMPLES:

sage: k5.<z> = CyclotomicField(5)
sage: k5.defining_polynomial()
x^4 + x^3 + x^2 + x + 1
sage: y = polygen(QQ,'y')
sage: k.<a> = NumberField(y^9 - 3*y + 5); k
Number Field in a with defining polynomial y^9 - 3*y + 5
sage: k.defining_polynomial()
y^9 - 3*y + 5

degree()¶

Return the degree of this number field.

EXAMPLES:

sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').degree()
3
sage: NumberField(x + 1, 'a').degree()
1
sage: NumberField(x^997 + 17*x + 3, 'a', check=False).degree()
997

different()¶

Compute the different fractional ideal of this number field.

The codifferent is the fractional ideal of all $x$ in $K$ such that the the trace of $xy$ is an integer for all $y \in O_K$ .

The different is the integral ideal which is the inverse of the codifferent.

EXAMPLES:

sage: k.<a> = NumberField(x^2 + 23)
sage: d = k.different()
sage: d        # random sign in output
Fractional ideal (-a)
sage: d.norm()
23
sage: k.disc()
-23

The different is cached:

sage: d is k.different()
True

Another example:

sage: k.<b> = NumberField(x^2 - 123)
sage: d = k.different(); d
Fractional ideal (2*b)
sage: d.norm()
492
sage: k.disc()
492

disc(v=None)¶

Shortcut for self.discriminant.

EXAMPLES:

sage: k.<b> = NumberField(x^2 - 123)
sage: k.disc()
492

discriminant(v=None)¶

Returns the discriminant of the ring of integers of the number field, or if v is specified, the determinant of the trace pairing on the elements of the list v.

INPUT:

v (optional) - list of element of this number field

OUTPUT: Integer if v is omitted, and Rational otherwise.

EXAMPLES:

sage: K.<t> = NumberField(x^3 + x^2 - 2*x + 8)
sage: K.disc()
-503
sage: K.disc([1, t, t^2])
-2012
sage: K.disc([1/7, (1/5)*t, (1/3)*t^2])
-2012/11025
sage: (5*7*3)^2
11025

elements_of_norm(n, proof=None)¶

Return a list of solutions modulo units of positive norm to $Norm(a) = n$ , where a can be any integer in this number field.

INPUT:

proof - default: True, unless you called number_field_proof and set it otherwise.

EXAMPLES:

sage: K.<a> = NumberField(x^2+1)
sage: K.elements_of_norm(3)
[]
sage: K.elements_of_norm(50)
[7*a - 1, -5*a + 5, a - 7]           # 32-bit
[7*a - 1, -5*a + 5, -7*a - 1]        # 64-bit

extension(poly, name=None, names=None, check=True, embedding=None)¶

Return the relative extension of this field by a given polynomial.

EXAMPLES:

sage: K.<a> = NumberField(x^3 - 2)
sage: R.<t> = K[]
sage: L.<b> = K.extension(t^2 + a); L
Number Field in b with defining polynomial t^2 + a over its base field

We create another extension.

sage: k.<a> = NumberField(x^2 + 1); k
Number Field in a with defining polynomial x^2 + 1
sage: y = var('y')
sage: m.<b> = k.extension(y^2 + 2); m
Number Field in b with defining polynomial y^2 + 2 over its base field

Note that b is a root of $y^2 + 2$ :

sage: b.minpoly()
x^2 + 2
sage: b.minpoly('z')
z^2 + 2

A relative extension of a relative extension.

sage: k.<a> = NumberField([x^2 + 1, x^3 + x + 1])
sage: R.<z> = k[]
sage: L.<b> = NumberField(z^3 + 3 + a); L
Number Field in b with defining polynomial z^3 + a0 + 3 over its base field

factor(n)¶

Ideal factorization of the principal ideal generated by $n$ .

EXAMPLE: Here we show how to factor Gaussian integers (up to units). First we form a number field defined by $x^2 + 1$ :

sage: K.<I> = NumberField(x^2 + 1); K
Number Field in I with defining polynomial x^2 + 1

Here are the factors:

sage: fi, fj = K.factor(13); fi,fj
((Fractional ideal (-3*I - 2), 1), (Fractional ideal (3*I - 2), 1))

Now we extract the reduced form of the generators:

sage: zi = fi[0].gens_reduced()[0]; zi
-3*I - 2
sage: zj = fj[0].gens_reduced()[0]; zj
3*I - 2

We recover the integer that was factored in $\ZZ[i]$ :

sage: zi*zj
13

We can see units are not considered:

sage: K.factor(2)
(Fractional ideal (I + 1))^2
sage: expand((I+1)^2)
2*I

One can also factor elements or ideals of the number field:

sage: K.<a> = NumberField(x^2 + 1)
sage: K.factor(1/3)
Fractional ideal (3)^-1
sage: K.factor(1+a)
Fractional ideal (a + 1)
sage: K.factor(1+a/5)
(Fractional ideal (-3*a - 2)) * (Fractional ideal (a + 1)) * (Fractional ideal (-a - 2))^-1 * (Fractional ideal (2*a + 1))^-1

The slightly odd syntax in the next example is to ensure the same result with both 32- and 64-bit systems.

sage: L.<b> = K.extension(x^2 - 7)
sage: list(L.factor(a + 1)) == [(L.ideal((a + b + 2)/2), 1), (L.ideal((a - b - 2)/2), 1)]
True

It doesn’t make sense to factor the ideal (0), so this raises an error:

sage: L.factor(0)
...
AttributeError: 'NumberFieldIdeal' object has no attribute 'factor'

AUTHORS:

Alex Clemesha (2006-05-20), Francis Clarke (2009-04-21): examples

fractional_ideal(*gens, **kwds)¶

Return the ideal in $\mathcal{O}_K$ generated by gens. This overrides the sage.rings.ring.Field method to use the sage.rings.ring.Ring one instead, since we’re not really concerned with ideals in a field but in its ring of integers.

INPUT:

gens - a list of generators, or a number field ideal.

EXAMPLES:

sage: K.<a> = NumberField(x^3-2)
sage: K.fractional_ideal([1/a])
Fractional ideal (1/2*a^2)

One can also input a number field ideal itself, or, more usefully, for a tower of number fields an ideal in one of the fields lower down the tower.

sage: K.fractional_ideal(K.ideal(a))
Fractional ideal (a)
sage: L.<b> = K.extension(x^2 - 3, x^2 + 1)
sage: M.<c> = L.extension(x^2 + 1)
sage: L.ideal(K.ideal(2, a))
Fractional ideal (a)
sage: M.ideal(K.ideal(2, a)) == M.ideal(a*(b - c)/2)
True

The zero ideal is not a fractional ideal!

sage: K.fractional_ideal(0)
...
ValueError: gens must have a nonzero element (zero ideal is not a fractional ideal)

galois_group(type=None, algorithm='pari', names=None)¶

Return the Galois group of the Galois closure of this number field.

INPUT:

type - none, gap, or pari. If None (the default), return an explicit group of automorphisms of self as a GaloisGroup_v2 object. Otherwise, return a GaloisGroup_v1 wrapper object based on a Pari or Gap transitive group object, which is quicker to compute, but rather less useful (in particular, it can’t be made to act on self). If type = ‘gap’, the database_gap package should be installed.
algorithm - ‘pari’, ‘kash’, ‘magma’. (default: ‘pari’, except when the degree is >= 12 when ‘kash’ is tried.)
name - a string giving a name for the generator of the Galois closure of self, when self is not Galois. This is ignored if type is not None.

Note that computing Galois groups as abstract groups is often much faster than computing them as explicit automorphism groups (but of course you get less information out!) For more (important!) documentation, so the documentation for Galois groups of polynomials over $\QQ$ , e.g., by typing K.polynomial().galois_group?, where $K$ is a number field.

To obtain actual field homomorphisms from the number field to its splitting field, use type=None.

EXAMPLES:

With type None:

sage: k.<b> = NumberField(x^2 - 14) # a Galois extension
sage: G = k.galois_group(); G
Galois group of Number Field in b with defining polynomial x^2 - 14
sage: G.gen(0)
(1,2)
sage: G.gen(0)(b)
-b
sage: G.artin_symbol(k.primes_above(3)[0])
(1,2)

sage: k.<b> = NumberField(x^3 - x + 1) # not Galois
sage: G = k.galois_group(names='c'); G
Galois group of Galois closure in c of Number Field in b with defining polynomial x^3 - x + 1
sage: G.gen(0)
(1,2)(3,5)(4,6)

With type 'pari':

sage: NumberField(x^3-2, 'a').galois_group(type="pari")
Galois group PARI group [6, -1, 2, "S3"] of degree 3 of the Number Field in a with defining polynomial x^3 - 2

sage: NumberField(x-1, 'a').galois_group(type="gap")    # optional - database_gap
Galois group Transitive group number 1 of degree 1 of the Number Field in a with defining polynomial x - 1
sage: NumberField(x^2+2, 'a').galois_group(type="gap")  # optional - database_gap
Galois group Transitive group number 1 of degree 2 of the Number Field in a with defining polynomial x^2 + 2
sage: NumberField(x^3-2, 'a').galois_group(type="gap")  # optional - database_gap
Galois group Transitive group number 2 of degree 3 of the Number Field in a with defining polynomial x^3 - 2

sage: x = polygen(QQ)
sage: NumberField(x^3 + 2*x + 1, 'a').galois_group(pari_group=False)    # optional - database_gap
Galois group Transitive group number 2 of degree 3 of the Number Field in a with defining polynomial x^3 + 2*x + 1            
sage: NumberField(x^3 + 2*x + 1, 'a').galois_group(algorithm='magma')   # optional - magma, , database_gap
Galois group Transitive group number 2 of degree 3 of the Number Field in a with defining polynomial x^3 + 2*x + 1

EXPLICIT GALOIS GROUP: We compute the Galois group as an explicit group of automorphisms of the Galois closure of a field.

sage: K.<a> = NumberField(x^3 - 2)
sage: L.<b1> = K.galois_closure(); L
Number Field in b1 with defining polynomial x^6 + 40*x^3 + 1372
sage: G = End(L); G
Automorphism group of Number Field in b1 with defining polynomial x^6 + 40*x^3 + 1372
sage: G.list()
[
Ring endomorphism of Number Field in b1 with defining polynomial x^6 + 40*x^3 + 1372
  Defn: b1 |--> b1,
...
Ring endomorphism of Number Field in b1 with defining polynomial x^6 + 40*x^3 + 1372
  Defn: b1 |--> -2/63*b1^4 - 31/63*b1
]
sage: G[1](b1)
1/36*b1^4 + 1/18*b1

gen(n=0)¶

Return the generator for this number field.

INPUT:

n - must be 0 (the default), or an exception is raised.

EXAMPLES:

sage: k.<theta> = NumberField(x^14 + 2); k
Number Field in theta with defining polynomial x^14 + 2
sage: k.gen()
theta
sage: k.gen(1)
...
IndexError: Only one generator.

gen_embedding()¶

If an embedding has been specified, return the image of the generator under that embedding. Otherwise return None.

EXAMPLES:

sage: QuadraticField(-7, 'a').gen_embedding()
2.645751311064591?*I
sage: NumberField(x^2+7, 'a').gen_embedding() # None

ideal(*gens, **kwds)¶

K.ideal() returns a fractional ideal of the field, except for the zero ideal which is not a fractional ideal.

EXAMPLES:

sage: K.<i>=NumberField(x^2+1)
sage: K.ideal(2)
Fractional ideal (2)
sage: K.ideal(2+i)
Fractional ideal (i + 2)
sage: K.ideal(0)
Ideal (0) of Number Field in i with defining polynomial x^2 + 1

ideals_of_bdd_norm(bound)¶

All integral ideals of bounded norm.

INPUT:

bound - a positive integer

OUTPUT: A dict of all integral ideals I such that Norm(I) <= bound, keyed by norm.

EXAMPLE:

sage: K.<a> = NumberField(x^2 + 23)
sage: d = K.ideals_of_bdd_norm(10)
sage: for n in d:
...       print n
...       for I in d[n]:
...           print I
1
Fractional ideal (1)
2
Fractional ideal (2, 1/2*a - 1/2)
Fractional ideal (2, 1/2*a + 1/2)
3
Fractional ideal (3, -1/2*a + 1/2)
Fractional ideal (3, -1/2*a - 1/2)
4
Fractional ideal (4, 1/2*a + 3/2)
Fractional ideal (2)
Fractional ideal (4, 1/2*a + 5/2)
5
6
Fractional ideal (-1/2*a + 1/2)
Fractional ideal (6, 1/2*a + 5/2)
Fractional ideal (6, 1/2*a + 7/2)
Fractional ideal (1/2*a + 1/2)
7
8
Fractional ideal (-1/2*a - 3/2)
Fractional ideal (4, a - 1)
Fractional ideal (4, a + 1)
Fractional ideal (1/2*a - 3/2)
9
Fractional ideal (9, 1/2*a + 11/2)
Fractional ideal (3)
Fractional ideal (9, 1/2*a + 7/2)
10

integral_basis(v=None)¶

Returns a list containing a ZZ-basis for the full ring of integers of this number field.

INPUT:

v - None, a prime, or a list of primes. See the documentation for self.maximal_order.

EXAMPLES:

sage: K.<a> = NumberField(x^5 + 10*x + 1)
sage: K.integral_basis()
[1, a, a^2, a^3, a^4]

Next we compute the ring of integers of a cubic field in which 2 is an “essential discriminant divisor”, so the ring of integers is not generated by a single element.

sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
sage: K.integral_basis()
[1, 1/2*a^2 + 1/2*a, a^2]

ALGORITHM: Uses the pari library.

is_absolute()¶

Returns True if self is an absolute field.

This function will be implemented in the derived classes.

EXAMPLES:

sage: K = CyclotomicField(5)
sage: K.is_absolute()
True

is_field(proof=True)¶

Return True since a number field is a field.

EXAMPLES:

sage: NumberField(x^5 + x + 3, 'c').is_field()
True

is_galois()¶

Return True if this number field is a Galois extension of $\QQ$ .

EXAMPLES:

sage: NumberField(x^2 + 1, 'i').is_galois()
True
sage: NumberField(x^3 + 2, 'a').is_galois()
False

is_isomorphic(other)¶

Return True if self is isomorphic as a number field to other.

EXAMPLES:

sage: k.<a> = NumberField(x^2 + 1)
sage: m.<b> = NumberField(x^2 + 4)
sage: k.is_isomorphic(m)
True
sage: m.<b> = NumberField(x^2 + 5)
sage: k.is_isomorphic (m)
False

sage: k = NumberField(x^3 + 2, 'a')
sage: k.is_isomorphic(NumberField((x+1/3)^3 + 2, 'b'))
True
sage: k.is_isomorphic(NumberField(x^3 + 4, 'b'))
True
sage: k.is_isomorphic(NumberField(x^3 + 5, 'b'))
False

is_relative()¶

EXAMPLES:

sage: K.<a> = NumberField(x^10 - 2)
sage: K.is_absolute()
True
sage: K.is_relative()
False

is_totally_imaginary()¶

Return True if self is totally imaginary, and False otherwise.

Totally imaginary means that no isomorphic embedding of self into the complex numbers has image contained in the real numbers.

EXAMPLES:

sage: NumberField(x^2+2, 'alpha').is_totally_imaginary()
True
sage: NumberField(x^2-2, 'alpha').is_totally_imaginary()
False
sage: NumberField(x^4-2, 'alpha').is_totally_imaginary()
False

is_totally_real()¶

Return True if self is totally real, and False otherwise.

Totally real means that every isomorphic embedding of self into the complex numbers has image contained in the real numbers.

EXAMPLES:

sage: NumberField(x^2+2, 'alpha').is_totally_real()
False
sage: NumberField(x^2-2, 'alpha').is_totally_real()
True
sage: NumberField(x^4-2, 'alpha').is_totally_real()
False

latex_variable_name(name=None)¶

Return the latex representation of the variable name for this number field.

EXAMPLES:

sage: NumberField(x^2 + 3, 'a').latex_variable_name()
'a'
sage: NumberField(x^3 + 3, 'theta3').latex_variable_name()
'\theta_{3}'
sage: CyclotomicField(5).latex_variable_name()
'\zeta_{5}'

narrow_class_group(proof=None)¶

Return the narrow class group of this field.

INPUT:

proof - default: None (use the global proof setting, which defaults to True).

EXAMPLES:

sage: NumberField(x^3+x+9, 'a').narrow_class_group()
Multiplicative Abelian Group isomorphic to C2

ngens()¶

Return the number of generators of this number field (always 1).

OUTPUT: the python integer 1.

EXAMPLES:

sage: NumberField(x^2 + 17,'a').ngens()
1
sage: NumberField(x + 3,'a').ngens()
1
sage: k.<a> = NumberField(x + 3)
sage: k.ngens()
1
sage: k.0
-3

number_of_roots_of_unity()¶

Return the number of roots of unity in this field.

Note

We do not create the full unit group since that can be expensive, but we do use it if it is already known.

EXAMPLES:

sage: F.<alpha> = NumberField(x**22+3)
sage: F.zeta_order()
6
sage: F.<alpha> = NumberField(x**2-7)
sage: F.zeta_order()
2

order()¶

Return the order of this number field (always +infinity).

OUTPUT: always positive infinity

EXAMPLES:

sage: NumberField(x^2 + 19,'a').order()
+Infinity

pari_bnf(certify=False, units=True)¶

PARI big number field corresponding to this field.

EXAMPLES:

sage: k.<a> = NumberField(x^2 + 1); k
Number Field in a with defining polynomial x^2 + 1
sage: len(k.pari_bnf())
10
sage: k.pari_bnf()[:4]
[[;], matrix(0,7), [;], ...]
sage: len(k.pari_nf())
9

pari_bnf_certify()¶

Run the PARI bnfcertify function to ensure the correctness of answers.

If this function returns True (and doesn’t raise a ValueError), then certification succeeded, and results that use the PARI bnf structure with this field are supposed to be correct.

Warning

I wouldn’t trust this to mean that everything computed involving this number field is actually correct.

EXAMPLES:

sage: k.<a> = NumberField(x^7 + 7); k
Number Field in a with defining polynomial x^7 + 7
sage: k.pari_bnf_certify()
True

pari_nf()¶

PARI number field corresponding to this field.

This is the number field constructed using nfinit. This is the same as the number field got by doing pari(self) or gp(self).

EXAMPLES:

sage: k.<a> = NumberField(x^4 - 3*x + 7); k
Number Field in a with defining polynomial x^4 - 3*x + 7
sage: k.pari_nf()[:4]
[x^4 - 3*x + 7, [0, 2], 85621, 1]
sage: pari(k)[:4]
[x^4 - 3*x + 7, [0, 2], 85621, 1]        

sage: k.<a> = NumberField(x^4 - 3/2*x + 5/3); k
Number Field in a with defining polynomial x^4 - 3/2*x + 5/3
sage: k.pari_nf()
...
TypeError: Unable to coerce number field defined by non-integral polynomial to PARI.
sage: pari(k)
...
TypeError: Unable to coerce number field defined by non-integral polynomial to PARI.
sage: gp(k)
...
TypeError: Unable to coerce number field defined by non-integral polynomial to PARI.

pari_polynomial(name='x')¶

PARI polynomial corresponding to polynomial that defines this field. By default, this is a polynomial in the variable “x”.

EXAMPLES:

sage: y = polygen(QQ)
sage: k.<a> = NumberField(y^2 - 3/2*y + 5/3)
sage: k.pari_polynomial()
x^2 - 3/2*x + 5/3
sage: k.pari_polynomial('a')
a^2 - 3/2*a + 5/3

polynomial()¶

Return the defining polynomial of this number field.

This is exactly the same as self.defining_polynomal().

EXAMPLES:

sage: NumberField(x^2 + (2/3)*x - 9/17,'a').polynomial()
x^2 + 2/3*x - 9/17

polynomial_ntl()¶

Return defining polynomial of this number field as a pair, an ntl polynomial and a denominator.

This is used mainly to implement some internal arithmetic.

EXAMPLES:

sage: NumberField(x^2 + (2/3)*x - 9/17,'a').polynomial_ntl()
([-27 34 51], 51)

polynomial_quotient_ring()¶

Return the polynomial quotient ring isomorphic to this number field.

EXAMPLES:

sage: K = NumberField(x^3 + 2*x - 5, 'alpha')
sage: K.polynomial_quotient_ring()
Univariate Quotient Polynomial Ring in alpha over Rational Field with modulus x^3 + 2*x - 5

polynomial_ring()¶

Return the polynomial ring that we view this number field as being a quotient of (by a principal ideal).

EXAMPLES: An example with an absolute field:

sage: k.<a> = NumberField(x^2 + 3)
sage: y = polygen(QQ, 'y')
sage: k.<a> = NumberField(y^2 + 3)
sage: k.polynomial_ring()
Univariate Polynomial Ring in y over Rational Field

An example with a relative field:

sage: y = polygen(QQ, 'y')
sage: M.<a> = NumberField([y^3 + 97, y^2 + 1]); M
Number Field in a0 with defining polynomial y^3 + 97 over its base field
sage: M.polynomial_ring()
Univariate Polynomial Ring in y over Number Field in a1 with defining polynomial y^2 + 1

power_basis()¶

Return a power basis for this number field over its base field.

If this number field is represented as $k[t]/f(t)$ , then the basis returned is $1, t, t^2, \ldots, t^{d-1}$ where $d$ is the degree of this number field over its base field.

EXAMPLES:

sage: K.<a> = NumberField(x^5 + 10*x + 1)
sage: K.power_basis()
[1, a, a^2, a^3, a^4]

sage: L.<b> = K.extension(x^2 - 2)
sage: L.power_basis()
[1, b]
sage: L.absolute_field('c').power_basis()
[1, c, c^2, c^3, c^4, c^5, c^6, c^7, c^8, c^9]

sage: M = CyclotomicField(15)
sage: M.power_basis()
[1, zeta15, zeta15^2, zeta15^3, zeta15^4, zeta15^5, zeta15^6, zeta15^7]

prime_above(x, degree=None)¶

Return a prime ideal of self lying over x.

INPUT:

x: usually an element or ideal of self. It should be such that self.ideal(x) is sensible. This excludes x=0.
degree (default: None): None or an integer. If one, find a prime above x of any degree. If an integer, find a prime above x such that the resulting residue field has exactly this degree.

OUTPUT: A prime ideal of self lying over x. If degree is specified and no such ideal exists, raises a ValueError.

EXAMPLES:

sage: x = ZZ['x'].gen()
sage: F.<t> = NumberField(x^3 - 2)

sage: P2 = F.prime_above(2)
sage: P2 # random
Fractional ideal (-t)
sage: 2 in P2
True
sage: P2.is_prime()
True
sage: P2.norm()
2

sage: P3 = F.prime_above(3)
sage: P3 # random
Fractional ideal (t + 1)
sage: 3 in P3
True
sage: P3.is_prime()
True
sage: P3.norm()
3

The ideal (3) is totally ramified in F, so there is no degree 2 prime above 3:

sage: F.prime_above(3, degree=2)
...
ValueError: No prime of degree 2 above Fractional ideal (3)
sage: [ id.residue_class_degree() for id, _ in F.ideal(3).factor() ]
[1]

Asking for a specific degree works:

sage: P5_1 = F.prime_above(5, degree=1)
sage: P5_1 # random
Fractional ideal (-t^2 - 1)
sage: P5_1.residue_class_degree()
1

sage: P5_2 = F.prime_above(5, degree=2)
sage: P5_2 # random
Fractional ideal (t^2 - 2*t - 1)
sage: P5_2.residue_class_degree()
2

Relative number fields are ok:

sage: G = F.extension(x^2 - 11, 'b')
sage: G.prime_above(7)
Fractional ideal (b + 2)

It doesn’t make sense to factor the ideal (0):

sage: F.prime_above(0)
...
AttributeError: 'NumberFieldIdeal' object has no attribute 'prime_factors'

prime_factors(x)¶

Return a list of the prime ideals of self which divide the ideal generated by $x$ .

OUTPUT: list of prime ideals (a new list is returned each time this function is called)

EXAMPLES:

sage: K.<w> = NumberField(x^2 + 23)
sage: K.prime_factors(w + 1)
[Fractional ideal (2, 1/2*w - 1/2),
Fractional ideal (2, 1/2*w + 1/2),
Fractional ideal (3, -1/2*w - 1/2)]

primes_above(x, degree=None)¶

Return prime ideals of self lying over x.

INPUT:

x: usually an element or ideal of self. It should be such that self.ideal(x) is sensible. This excludes x=0.
degree (default: None): None or an integer. If None, find all primes above x of any degree. If an integer, find all primes above x such that the resulting residue field has exactly this degree.

OUTPUT: A list of prime ideals of self lying over x. If degree is specified and no such ideal exists, returns the empty list. The output is sorted by residue degree first, then by underlying prime (or equivalently, by norm).

EXAMPLES:

sage: x = ZZ['x'].gen()
sage: F.<t> = NumberField(x^3 - 2)

sage: P2s = F.primes_above(2)
sage: P2s # random
[Fractional ideal (-t)]
sage: all(2 in P2 for P2 in P2s)
True
sage: all(P2.is_prime() for P2 in P2s)
True
sage: [ P2.norm() for P2 in P2s ]
[2]

sage: P3s = F.primes_above(3)
sage: P3s # random
[Fractional ideal (t + 1)]
sage: all(3 in P3 for P3 in P3s)
True
sage: all(P3.is_prime() for P3 in P3s)
True
sage: [ P3.norm() for P3 in P3s ]
[3]

The ideal (3) is totally ramified in F, so there is no degree 2 prime above 3:

sage: F.primes_above(3, degree=2)
[]
sage: [ id.residue_class_degree() for id, _ in F.ideal(3).factor() ]
[1]

Asking for a specific degree works:

sage: P5_1s = F.primes_above(5, degree=1)
sage: P5_1s # random
[Fractional ideal (-t^2 - 1)]
sage: P5_1 = P5_1s[0]; P5_1.residue_class_degree()
1

sage: P5_2s = F.primes_above(5, degree=2)
sage: P5_2s # random
[Fractional ideal (t^2 - 2*t - 1)]
sage: P5_2 = P5_2s[0]; P5_2.residue_class_degree()
2

Works in relative extensions too:

sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: I = F.ideal(a + 2*b)
sage: P, Q = K.primes_above(I)
sage: K.ideal(I) == P^4*Q
True
sage: K.primes_above(I, degree=1) == [P]
True
sage: K.primes_above(I, degree=4) == [Q]
True

It doesn’t make sense to factor the ideal (0), so this raises an error:

sage: F.prime_above(0)
...
AttributeError: 'NumberFieldIdeal' object has no attribute 'prime_factors'

primes_of_degree_one_iter(num_integer_primes=10000, max_iterations=100)¶

Return an iterator yielding prime ideals of absolute degree one and small norm.

Warning

It is possible that there are no primes of $K$ of absolute degree one of small prime norm, and it possible that this algorithm will not find any primes of small norm.

See module sage.rings.number_field.small_primes_of_degree_one for details.

INPUT:

num_integer_primes (default: 10000) - an integer. We try to find primes of absolute norm no greater than the num_integer_primes-th prime number. For example, if num_integer_primes is 2, the largest norm found will be 3, since the second prime is 3.
max_iterations (default: 100) - an integer. We test max_iterations integers to find small primes before raising StopIteration.

EXAMPLES:

sage: K.<z> = CyclotomicField(10)
sage: it = K.primes_of_degree_one_iter()
sage: Ps = [ it.next() for i in range(3) ]
sage: Ps # random
[Fractional ideal (z^3 + z + 1), Fractional ideal (3*z^3 - z^2 + z - 1), Fractional ideal (2*z^3 - 3*z^2 + z - 2)]
sage: [ P.norm() for P in Ps ] # random
[11, 31, 41]
sage: [ P.residue_class_degree() for P in Ps ]
[1, 1, 1]

primes_of_degree_one_list(n, num_integer_primes=10000, max_iterations=100)¶

Return a list of n prime ideals of absolute degree one and small norm.

Warning

It is possible that there are no primes of $K$ of absolute degree one of small prime norm, and it possible that this algorithm will not find any primes of small norm.

See module sage.rings.number_field.small_primes_of_degree_one for details.

INPUT:

num_integer_primes (default: 10000) - an integer. We try to find primes of absolute norm no greater than the num_integer_primes-th prime number. For example, if num_integer_primes is 2, the largest norm found will be 3, since the second prime is 3.
max_iterations (default: 100) - an integer. We test max_iterations integers to find small primes before raising StopIteration.

EXAMPLES:

sage: K.<z> = CyclotomicField(10)
sage: Ps = K.primes_of_degree_one_list(3)
sage: Ps
[Fractional ideal (z^3 + z + 1), Fractional ideal (3*z^3 - z^2 + z - 1), Fractional ideal (2*z^3 - 3*z^2 + z - 2)]
sage: [ P.norm() for P in Ps ]
[11, 31, 41]
sage: [ P.residue_class_degree() for P in Ps ]
[1, 1, 1]

primitive_element()¶

Return a primitive element for this field, i.e., an element that generates it over $\QQ$ .

EXAMPLES:

sage: K.<a> = NumberField(x^3 + 2)
sage: K.primitive_element()
a
sage: K.<a,b,c> = NumberField([x^2-2,x^2-3,x^2-5])
sage: K.primitive_element()
a - b + c
sage: alpha = K.primitive_element(); alpha
a - b + c
sage: alpha.minpoly()
x^2 + (2*b - 2*c)*x - 2*c*b + 6
sage: alpha.absolute_minpoly()
x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576

primitive_root_of_unity()¶

Return a generator of the roots of unity in this field.

Note

We do not create the full unit group since that can be expensive, but we do use it if it is already known.

EXAMPLES:

sage: K.<i> = NumberField(x^2+1)
sage: z = K.primitive_root_of_unity(); z
i
sage: z.multiplicative_order()
4

sage: K.<a> = NumberField(x^2+x+1)
sage: z = K.primitive_root_of_unity(); z
-a
sage: z.multiplicative_order()
6

random_element(num_bound=None, den_bound=None, integral_coefficients=False, distribution=None)¶

Return a random element of this number field.

INPUT:

num_bound - Bound on numerator of the coefficients of

the resulting element
den_bound - Bound on denominators of the coefficients

of the resulting element
integral_coefficients (default: False) - If True, then

the resulting element will have integral coefficients. This option overrides any value of $den_bound$ .
distribution - Distribution to use for the coefficients

of the resulting element

OUTPUT:

Element of this number field

EXAMPLES:

sage: K.<j> = NumberField(x^8+1)
sage: K.random_element()
1/2*j^7 - j^6 - 12*j^5 + 1/2*j^4 - 1/95*j^3 - 1/2*j^2 - 4

sage: K.<a,b,c> = NumberField([x^2-2,x^2-3,x^2-5])
sage: K.random_element()
((6136*c - 7489/3)*b + 5825/3*c - 71422/3)*a + (-4849/3*c + 58918/3)*b - 45718/3*c + 75409/12

sage: K.<a> = NumberField(x^5-2)
sage: K.random_element(integral_coefficients=True)
a^3 + a^2 - 3*a - 1

real_embeddings(prec=53)¶

Return all homomorphisms of this number field into the approximate real field with precision prec.

If prec is 53 (the default), then the real double field is used; otherwise the arbitrary precision (but slow) real field is used. If you want embeddings into the 53-bit double precision, which is faster, use self.embeddings(RDF).

EXAMPLES:

sage: K.<a> = NumberField(x^3 + 2)
sage: K.real_embeddings()
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Real Field with 53 bits of precision
  Defn: a |--> -1.25992104989487
]
sage: K.real_embeddings(16)
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Real Field with 16 bits of precision
  Defn: a |--> -1.260
]
sage: K.real_embeddings(100)
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Real Field with 100 bits of precision
  Defn: a |--> -1.2599210498948731647672106073
]

reduced_basis(prec=None)¶

This function returns an LLL-reduced basis for the Minkowski-embedding of the maximal order of a number field.

INPUT:

self - number field, the base field
prec (default: None) - the precision with which to compute the Minkowski embedding. (See NOTE below.)

OUTPUT:

An LLL-reduced basis for the Minkowski-embedding of the maximal order of a number field, given by a sequence of (integral) elements from the field.

Note

In the non-totally-real case, the LLL routine we call is currently Pari’s qflll(), which works with floating point approximations, and so the result is only as good as the precision promised by Pari. The matrix returned will always be integral; however, it may only be only “almost” LLL-reduced when the precision is not sufficiently high.

If the following run-time error occurs: “PariError: not a definite matrix in lllgram (42)” try increasing the prec parameter,

EXAMPLES:

sage: F.<t> = NumberField(x^6-7*x^4-x^3+11*x^2+x-1)
sage: F.maximal_order().basis()
[1/2*t^5 + 1/2*t^4 + 1/2*t^2 + 1/2, t, t^2, t^3, t^4, t^5]
sage: F.reduced_basis()
[-1, -1/2*t^5 + 1/2*t^4 + 3*t^3 - 3/2*t^2 - 4*t - 1/2, t, 1/2*t^5 + 1/2*t^4 - 4*t^3 - 5/2*t^2 + 7*t + 1/2, 1/2*t^5 - 1/2*t^4 - 2*t^3 + 3/2*t^2 - 1/2, 1/2*t^5 - 1/2*t^4 - 3*t^3 + 5/2*t^2 + 4*t - 5/2]
sage: CyclotomicField(12).reduced_basis()
[1, zeta12^2, zeta12, zeta12^3]

sage: F.<alpha> = NumberField(x^4+x^2+712312*x+131001238)
sage: F.integral_basis()
[1, alpha, 1/2*alpha^3 + 1/2*alpha^2, alpha^3]
sage: F.reduced_basis(prec=64)
[1, alpha, alpha^3 - 2*alpha^2 + 15*alpha, 16*alpha^3 - 31*alpha^2 + 469*alpha + 267109]     # 32-bit
...                                                # 64-bit
PariError: not a definite matrix in lllgram (42)   # 64-bit
sage: F.reduced_basis(prec=96)
[1, alpha, alpha^3 - 2*alpha^2 + 15*alpha, 16*alpha^3 - 31*alpha^2 + 469*alpha + 267109]

reduced_gram_matrix(prec=None)¶

This function returns the Gram matrix of an LLL-reduced basis for the Minkowski embedding of the maximal order of a number field.

INPUT:

self - number field, the base field
prec (default: None) - the precision with which to calculate the Minkowski embedding. (See NOTE below.)

OUTPUT: The Gram matrix $[<x_i,x_j>]$ of an LLL reduced basis for the maximal order of self, where the integral basis for self is given by $\{x_0, \dots, x_{n-1}\}$ . Here < , > is the usual inner product on $\RR^n$ , and self is embedded in $\RR^n$ by the Minkowski embedding. See the docstring for Minkowski_embedding() for more information.

Note

In the non-totally-real case, the LLL routine we call is currently Pari’s qflll(), which works with floating point approximations, and so the result is only as good as the precision promised by Pari. In particular, in this case, the returned matrix will not be integral, and may not have enough precision to recover the correct gram matrix (which is known to be integral for theoretical reasons). Thus the need for the prec flag above.

If the following run-time error occurs: “PariError: not a definite matrix in lllgram (42)” try increasing the prec parameter,

EXAMPLES:

sage: F.<t> = NumberField(x^6-7*x^4-x^3+11*x^2+x-1)
sage: F.reduced_gram_matrix()
[ 6  3  0  2  0  1]
[ 3  9  0  1  0 -2]
[ 0  0 14  6 -2  3]
[ 2  1  6 16 -3  3]
[ 0  0 -2 -3 16  6]
[ 1 -2  3  3  6 19]
sage: Matrix(6, [(x*y).trace() for x in F.integral_basis() for y in F.integral_basis()])
[2550  133  259  664 1368 3421]
[ 133   14    3   54   30  233]
[ 259    3   54   30  233  217]
[ 664   54   30  233  217 1078]
[1368   30  233  217 1078 1371]
[3421  233  217 1078 1371 5224]

sage: var('x')
x
sage: F.<alpha> = NumberField(x^4+x^2+712312*x+131001238)
sage: F.reduced_gram_matrix(prec=128)
[   4.0000000000000000000000000000000000000   0.00000000000000000000000000000000000000 -2.1369320000000000000000000000000000000e6 -3.3122478000000000000000000000000000000e7]
[  0.00000000000000000000000000000000000000    46721.539331563218381658483353092335550 -2.2467769057394530109094755223395819322e7 -3.4807276041138450473611629088647496430e8]
[-2.1369320000000000000000000000000000000e6 -2.2467769057394530109094755223395819322e7 7.0704243186034907491782135494859351061e12 1.1256636615786237006027526953641297995e14]
[-3.3122478000000000000000000000000000000e7 -3.4807276041138450473611629088647496430e8 1.1256636615786237006027526953641297995e14 1.7923838231014970520503146603069479547e15]

regulator(proof=None)¶

Return the regulator of this number field.

Note that PARI computes the regulator to higher precision than the Sage default.

INPUT:

proof - default: True, unless you set it otherwise.

EXAMPLES:

sage: NumberField(x^2-2, 'a').regulator()
0.881373587019543
sage: NumberField(x^4+x^3+x^2+x+1, 'a').regulator()
0.962423650119207

residue_field(prime, names=None, check=True)¶

Return the residue field of this number field at a given prime, ie $O_K / p O_K$ .

INPUT:

prime - a prime ideal of the maximal order in this number field, or an element of the field which generates a principal prime ideal.
names - the name of the variable in the residue field
check - whether or not to check the primality of prime.

OUTPUT: The residue field at this prime.

EXAMPLES:

sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^4+3*x^2-17)
sage: P = K.ideal(61).factor()[0][0]
sage: K.residue_field(P)
Residue field in abar of Fractional ideal (-2*a^2 + 1)

sage: K.<i> = NumberField(x^2 + 1)
sage: K.residue_field(1+i)
Residue field of Fractional ideal (i + 1)

TESTS:

sage: L.<b> = NumberField(x^2 + 5)
sage: L.residue_field(P)
...
ValueError: Fractional ideal (-2*a^2 + 1) is not an ideal of Number Field in b with defining polynomial x^2 + 5
sage: L.residue_field(2)
...
ValueError: Fractional ideal (2) is not a prime ideal

sage: L.residue_field(L.prime_above(5)^2)
...
ValueError: Fractional ideal (5) is not a prime ideal

roots_of_unity()¶

Return all the roots of unity in this field, primitive or not.

EXAMPLES:

sage: K.<b> = NumberField(x^2+1)
sage: zs = K.roots_of_unity(); zs
[b, -1, -b, 1]
sage: [ z**K.number_of_roots_of_unity() for z in zs ]
[1, 1, 1, 1]

selmer_group(S, m, proof=True)¶

Compute the Selmer group $K(S,m)$ , which is defined to be the subgroup of $K^\times/(K^\times)^m$ consisting of elements $a$ such that $K(\sqrt[m]{a})/K$ is unramified at all primes of $K$ lying above a place outside of $S$ .

INPUT:

S - A set of primes of self.
m - A positive integer.
proof - If False, assume Pari’s GRH++ in computing the class group.

OUTPUT:

A list of generators of $K(S,m)$ .

EXAMPLES:

sage: K.<a> = QuadraticField(-5)
sage: K.selmer_group((), 2)
[-1, 2]
sage: K.selmer_group([K.ideal(2, -a+1)], 2)
[2, -1]
sage: K.selmer_group([K.ideal(2, -a+1), K.ideal(3, a+1)], 2)
[2, a + 1, -1]
sage: K.selmer_group((K.ideal(2, -a+1),K.ideal(3, a+1)), 4)
[2, a + 1, -1]
sage: K.selmer_group([K.ideal(2, -a+1)], 3)
[2]
sage: K.selmer_group([K.ideal(2, -a+1), K.ideal(3, a+1)], 3)
[2, a + 1]
sage: K.selmer_group([K.ideal(2, -a+1), K.ideal(3, a+1), K.ideal(a)], 3)
[2, a + 1, -a]
sage: K.<a> = NumberField(polygen(QQ))
sage: K.selmer_group([],5)
[]

signature()¶

Return (r1, r2), where r1 and r2 are the number of real embeddings and pairs of complex embeddings of this field, respectively.

EXAMPLES:

sage: NumberField(x^2+1, 'a').signature()
(0, 1)
sage: NumberField(x^3-2, 'a').signature()
(1, 1)

specified_complex_embedding()¶

Returns the embedding of this field into the complex numbers which has been specified.

Fields created with the QuadraticField or CyclotomicField constructors come with an implicit embedding. To get one of these fields without the embedding, use the generic NumberField constructor.

EXAMPLES:

sage: QuadraticField(-1, 'I').specified_complex_embedding()
Generic morphism:
  From: Number Field in I with defining polynomial x^2 + 1
  To:   Complex Lazy Field
  Defn: I -> 1*I

sage: QuadraticField(3, 'a').specified_complex_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^2 - 3
  To:   Real Lazy Field
  Defn: a -> 1.732050807568878?

sage: CyclotomicField(13).specified_complex_embedding()
Generic morphism:
  From: Cyclotomic Field of order 13 and degree 12
  To:   Complex Lazy Field
  Defn: zeta13 -> 0.885456025653210? + 0.464723172043769?*I

Most fields don’t implicitly have embeddings unless explicitly specified:

sage: NumberField(x^2-2, 'a').specified_complex_embedding() is None
True
sage: NumberField(x^3-x+5, 'a').specified_complex_embedding() is None
True
sage: NumberField(x^3-x+5, 'a', embedding=2).specified_complex_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^3 - x + 5
  To:   Real Lazy Field
  Defn: a -> -1.904160859134921?
sage: NumberField(x^3-x+5, 'a', embedding=CDF.0).specified_complex_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^3 - x + 5
  To:   Complex Lazy Field
  Defn: a -> 0.952080429567461? + 1.311248044077123?*I

This function only returns complex embeddings:

sage: K.<a> = NumberField(x^2-2, embedding=Qp(7)(2).sqrt())
sage: K.specified_complex_embedding() is None
True
sage: K.gen_embedding()
3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8 + 6*7^9 + 6*7^10 + 2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16 + 7^17 + 4*7^18 + 6*7^19 + O(7^20)
sage: K.coerce_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^2 - 2
  To:   7-adic Field with capped relative precision 20
  Defn: a -> 3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8 + 6*7^9 + 6*7^10 + 2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16 + 7^17 + 4*7^18 + 6*7^19 + O(7^20)

structure()¶

Return fixed isomorphism or embedding structure on self.

This is used to record various isomorphisms or embeddings that arise naturally in other constructions.

EXAMPLES:

sage: K.<z> = NumberField(x^2 + 3)
sage: L.<a> = K.absolute_field(); L
Number Field in a with defining polynomial x^2 + 3
sage: L.structure()
(Isomorphism given by variable name change map:
  From: Number Field in a with defining polynomial x^2 + 3
  To:   Number Field in z with defining polynomial x^2 + 3,
 Isomorphism given by variable name change map:
  From: Number Field in z with defining polynomial x^2 + 3
  To:   Number Field in a with defining polynomial x^2 + 3)

sage: K.<a> = QuadraticField(-3) 
sage: R.<y> = K[] 
sage: D.<x0> = K.extension(y) 
sage: D_abs.<y0> = D.absolute_field() 
sage: D_abs.structure()[0](y0) 
-a 

subfield(alpha, name=None, names=None)¶

Return an absolute number field K isomorphic to QQ(alpha) and a map from K to self that sends the generator of K to alpha.

INPUT:

alpha - an element of self, or something that coerces to an element of self.

OUTPUT:

K - a number field
from_K - a homomorphism from K to self that sends the generator of K to alpha.

EXAMPLES:

sage: K.<a> = NumberField(x^4 - 3); K
Number Field in a with defining polynomial x^4 - 3
sage: H.<b>, from_H = K.subfield(a^2)
sage: H
Number Field in b with defining polynomial x^2 - 3
sage: from_H(b)
a^2
sage: from_H
Ring morphism:
  From: Number Field in b with defining polynomial x^2 - 3
  To:   Number Field in a with defining polynomial x^4 - 3
  Defn: b |--> a^2

sage: K.<z> = CyclotomicField(5)
sage: K.subfield(z-z^2-z^3+z^4)
(Number Field in z0 with defining polynomial x^2 - 5,
Ring morphism:
From: Number Field in z0 with defining polynomial x^2 - 5
To:   Cyclotomic Field of order 5 and degree 4
Defn: z0 |--> -2*z^3 - 2*z^2 - 1)

You can also view a number field as having a different generator by just choosing the input to generate the whole field; for that it is better to use self.change_generator, which gives isomorphisms in both directions.

trace_pairing(v)¶

Return the matrix of the trace pairing on the elements of the list $v$ .

EXAMPLES:

sage: K.<zeta3> = NumberField(x^2 + 3)
sage: K.trace_pairing([1,zeta3])
[ 2  0]
[ 0 -6]

uniformizer(P, others='positive')¶

Returns an element of self with valuation 1 at the prime ideal P.

INPUT:

self - a number field
P - a prime ideal of self
others - either “positive” (default), in which case the element will have non-negative valuation at all other primes of self, or “negative”, in which case the element will have non-positive valuation at all other primes of self.

Note

When P is principal (e.g. always when self has class number one) the result may or may not be a generator of P!

EXAMPLES:

sage: K.<a> = NumberField(x^2 + 5); K
Number Field in a with defining polynomial x^2 + 5
sage: P,Q = K.ideal(3).prime_factors()
sage: P
Fractional ideal (3, a + 1)
sage: pi=K.uniformizer(P); pi
a + 1
sage: K.ideal(pi).factor()
(Fractional ideal (2, a + 1)) * (Fractional ideal (3, a + 1))
sage: pi=K.uniformizer(P,'negative'); pi
1/2*a + 1/2
sage: K.ideal(pi).factor()
(Fractional ideal (2, a + 1))^-1 * (Fractional ideal (3, a + 1))

sage: K = CyclotomicField(9)
sage: Plist=K.ideal(17).prime_factors()
sage: pilist = [K.uniformizer(P) for P in Plist]
sage: [pi.is_integral() for pi in pilist]
[True, True, True]
sage: [pi.valuation(P) for pi,P in zip(pilist,Plist)]
[1, 1, 1]
sage: [ pilist[i] in Plist[i] for i in range(len(Plist)) ]
[True, True, True]

unit_group(proof=None)¶

Return the unit group (including torsion) of this number field.

ALGORITHM: Uses PARI’s bnfunit command.

INPUT:

proof (bool, default True) flag passed to pari.

Note

The group is cached.

EXAMPLES:

sage: x = QQ['x'].0
sage: A = x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3
sage: K = NumberField(A, 'a')
sage: U = K.unit_group(); U
Unit group with structure C10 x Z of Number Field in a with defining polynomial x^4 - 10*x^3 + 100*x^2 - 375*x + 1375
sage: U.gens()
[-1/275*a^3 + 7/55*a^2 - 6/11*a + 4, 8/275*a^3 - 12/55*a^2 + 15/11*a - 2]
sage: U.invariants()
[10, 0]
sage: [u.multiplicative_order() for u in U.gens()]
[10, +Infinity]

Sage might not be able to provably compute the unit group:

sage: K = NumberField(x^17 + 3, 'a')
sage: K.unit_group(proof=True) # default
...
PariError: not enough precomputed primes, need primelimit ~  (35)

In this case, one can ask for the conjectural unit group (correct if the Generalized Riemann Hypothesis is true):

sage: U = K.unit_group(proof=False); U; U.gens()
Unit group with structure C2 x Z x Z x Z x Z x Z x Z x Z x Z of Number Field in a with defining polynomial x^17 + 3
[-1,
a^9 + a - 1,
a^16 - a^15 + a^14 - a^12 + a^11 - a^10 - a^8 + a^7 - 2*a^6 + a^4 - 3*a^3 + 2*a^2 - 2*a + 1,
2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4,
2*a^16 - 3*a^15 + 3*a^14 - 3*a^13 + 3*a^12 - a^11 + a^9 - 3*a^8 + 4*a^7 - 5*a^6 + 6*a^5 - 4*a^4 + 3*a^3 - 2*a^2 - 2*a + 4,
a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2,
a^16 - a^15 - 3*a^14 - 4*a^13 - 4*a^12 - 3*a^11 - a^10 + 2*a^9 + 4*a^8 + 5*a^7 + 4*a^6 + 2*a^5 - 2*a^4 - 6*a^3 - 9*a^2 - 9*a - 7,
a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1,
3*a^16 + 3*a^15 + 3*a^14 + 3*a^13 + 3*a^12 + 2*a^11 + 2*a^10 + 2*a^9 + a^8 - a^7 - 2*a^6 - 3*a^5 - 3*a^4 - 4*a^3 - 6*a^2 - 8*a - 8]

The provable and the conjectural results are cached separately (this fixes trac #2504):

sage: K.units(proof=True)
...
PariError: not enough precomputed primes, need primelimit ~  (35)

units(proof=None)¶

Return generators for the unit group modulo torsion.

INPUT:

proof (bool, default True) flag passed to pari.

Note

For more functionality see the unit_group() function.

ALGORITHM: Uses PARI’s bnfunit command.

EXAMPLES:

sage: x = QQ['x'].0
sage: A = x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3
sage: K = NumberField(A, 'a')
sage: K.units()
[8/275*a^3 - 12/55*a^2 + 15/11*a - 2]

Sage might not be able to provably compute the unit group:

sage: K = NumberField(x^17 + 3, 'a')
sage: K.units(proof=True) # default
...
PariError: not enough precomputed primes, need primelimit ~  (35)

In this case, one can ask for the conjectural unit group (correct if the Generalized Riemann Hypothesis is true):

sage: K.units(proof=False)
[a^9 + a - 1,
a^16 - a^15 + a^14 - a^12 + a^11 - a^10 - a^8 + a^7 - 2*a^6 + a^4 - 3*a^3 + 2*a^2 - 2*a + 1,
2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4,
2*a^16 - 3*a^15 + 3*a^14 - 3*a^13 + 3*a^12 - a^11 + a^9 - 3*a^8 + 4*a^7 - 5*a^6 + 6*a^5 - 4*a^4 + 3*a^3 - 2*a^2 - 2*a + 4,
a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2,
a^16 - a^15 - 3*a^14 - 4*a^13 - 4*a^12 - 3*a^11 - a^10 + 2*a^9 + 4*a^8 + 5*a^7 + 4*a^6 + 2*a^5 - 2*a^4 - 6*a^3 - 9*a^2 - 9*a - 7,
a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1,
3*a^16 + 3*a^15 + 3*a^14 + 3*a^13 + 3*a^12 + 2*a^11 + 2*a^10 + 2*a^9 + a^8 - a^7 - 2*a^6 - 3*a^5 - 3*a^4 - 4*a^3 - 6*a^2 - 8*a - 8]

The provable and the conjectural results are cached separately (this fixes trac #2504):

sage: K.units(proof=True)
...
PariError: not enough precomputed primes, need primelimit ~  (35)

zeta(n=2, all=False)¶

Return one, or a list of all, primitive n-th root of unity in this field.

INPUT:

n - positive integer
all - bool. If False (default), return a primitive $n$ -th root of unity in this field, or raise a ValueError exception if there are none. If True, return a list of all primitive $n$ -th roots of unity in this field (possibly empty).

Note

To obtain the maximal order of a root of unity in this field, use self.number_of_roots_of_unity().

Note

We do not create the full unit group since that can be expensive, but we do use it if it is already known.

EXAMPLES:

sage: K.<z> = NumberField(x^2 + 3)
sage: K.zeta(1)
1
sage: K.zeta(2)
-1
sage: K.zeta(2, all=True)
[-1]
sage: K.zeta(3)
1/2*z - 1/2            
sage: K.zeta(3, all=True)
[1/2*z - 1/2, -1/2*z - 1/2]
sage: K.zeta(4)
...
ValueError: There are no 4th roots of unity in self.

sage: r.<x> = QQ[]
sage: K.<b> = NumberField(x^2+1)
sage: K.zeta(4)
b
sage: K.zeta(4,all=True)
[b, -b]
sage: K.zeta(3)
...
ValueError: There are no 3rd roots of unity in self.
sage: K.zeta(3,all=True)
[]

zeta_coefficients(n)¶

Compute the first n coefficients of the Dedekind zeta function of this field as a Dirichlet series.

EXAMPLE:

sage: x = QQ['x'].0
sage: NumberField(x^2+1, 'a').zeta_coefficients(10)
[1, 1, 0, 1, 2, 0, 0, 1, 1, 2]

zeta_function(prec=53, max_imaginary_part=0, max_asymp_coeffs=40)¶

Return the Zeta function of this number field.

This actually returns an interface to Tim Dokchitser’s program for computing with the Dedekind zeta function zeta_F(s) of the number field F.

INPUT:

prec - integer (bits precision)
max_imaginary_part - real number
max_asymp_coeffs - integer

OUTPUT: The zeta function of this number field.

EXAMPLES:

sage: K.<a> = NumberField(ZZ['x'].0^2+ZZ['x'].0-1)
sage: Z = K.zeta_function()
sage: Z
Zeta function associated to Number Field in a with defining polynomial x^2 + x - 1
sage: Z(-1)
0.0333333333333333
sage: L.<a, b, c> = NumberField([x^2 - 5, x^2 + 3, x^2 + 1])
sage: Z = L.zeta_function()
sage: Z(5)
1.00199015670185

zeta_order()¶

Return the number of roots of unity in this field.

Note

We do not create the full unit group since that can be expensive, but we do use it if it is already known.

EXAMPLES:

sage: F.<alpha> = NumberField(x**22+3)
sage: F.zeta_order()
6
sage: F.<alpha> = NumberField(x**2-7)
sage: F.zeta_order()
2

sage.rings.number_field.number_field.NumberField_generic_v1(poly, name, latex_name, canonical_embedding=None)¶

This is used in pickling generic number fields.

EXAMPLES:

sage: from sage.rings.number_field.number_field import NumberField_generic_v1
sage: R.<x> = QQ[]
sage: NumberField_generic_v1(x^2 + 1, 'i', 'i')
Number Field in i with defining polynomial x^2 + 1

class sage.rings.number_field.number_field.NumberField_quadratic(polynomial, name=None, latex_name=None, check=True, embedding=None)¶

Bases: sage.rings.number_field.number_field.NumberField_absolute

Create a quadratic extension of the rational field.

The command QuadraticField(a) creates the field $\QQ(\sqrt{a})$ .

EXAMPLES:

sage: QuadraticField(3, 'a')
Number Field in a with defining polynomial x^2 - 3
sage: QuadraticField(-4, 'b')
Number Field in b with defining polynomial x^2 + 4

class_number(proof=None)¶

Return the size of the class group of self.

If proof = False (not the default!) and the discriminant of the field is negative, then the following warning from the PARI manual applies:

Warning

For $D<0$ , this function may give incorrect results when the class group has a low exponent (has many cyclic factors), because implementing Shank’s method in full generality slows it down immensely.

EXAMPLES:

sage: QuadraticField(-23,'a').class_number()
3

These are all the primes so that the class number of $\QQ(\sqrt{-p})$ is $1$ :

sage: [d for d in prime_range(2,300) if not is_square(d) and QuadraticField(-d,'a').class_number() == 1]
[2, 3, 7, 11, 19, 43, 67, 163]

It is an open problem to prove that there are infinity many positive square-free $d$ such that $\QQ(\sqrt{d})$ has class number $1$ :

sage: len([d for d in range(2,200) if not is_square(d) and QuadraticField(d,'a').class_number() == 1])
121

TESTS:

sage: type(QuadraticField(-23,'a').class_number())
<type 'sage.rings.integer.Integer'>
sage: type(NumberField(x^3 + 23, 'a').class_number())
<type 'sage.rings.integer.Integer'>
sage: type(NumberField(x^3 + 23, 'a').extension(x^2 + 5, 'b').class_number())
<type 'sage.rings.integer.Integer'>
sage: type(CyclotomicField(10).class_number())
<type 'sage.rings.integer.Integer'>

discriminant(v=None)¶

Returns the discriminant of the ring of integers of the number field, or if v is specified, the determinant of the trace pairing on the elements of the list v.

INPUT:

v (optional) - list of element of this number field

OUTPUT: Integer if v is omitted, and Rational otherwise.

EXAMPLES:

sage: K.<i> = NumberField(x^2+1)
sage: K.discriminant()
-4
sage: K.<a> = NumberField(x^2+5)
sage: K.discriminant()
-20
sage: K.<a> = NumberField(x^2-5)
sage: K.discriminant()
5

hilbert_class_field(names)¶

Returns the Hilbert class field of this quadratic field as a relative extension of this field.

Note

For the polynomial that defines this field as a relative extension, see the hilbert_class_field_defining_polynomial command, which is vastly faster than this command, since it doesn’t construct a relative extension.

EXAMPLES:

sage: K.<a> = NumberField(x^2 + 23)
sage: L = K.hilbert_class_field('b'); L
Number Field in b with defining polynomial x^3 + x^2 - 1 over its base field
sage: L.absolute_field('c')
Number Field in c with defining polynomial x^6 + 2*x^5 + 70*x^4 + 90*x^3 + 1631*x^2 + 1196*x + 12743
sage: K.hilbert_class_field_defining_polynomial()
x^3 + x^2 - 1

hilbert_class_field_defining_polynomial(name='x')¶

Returns a polynomial over $\QQ$ whose roots generate Hilbert class field of this quadratic field as an extension of this quadratic field.

Note

Computed using PARI via Schertz’s method. This implementation is quite fast.

EXAMPLES:

sage: K.<b> = QuadraticField(-23)
sage: K.hilbert_class_field_defining_polynomial()
x^3 + x^2 - 1

Note that this polynomial is not the actual Hilbert class polynomial: see hilbert_class_polynomial.

sage: K.hilbert_class_polynomial()
x^3 + 3491750*x^2 - 5151296875*x + 12771880859375

sage: K.<a> = QuadraticField(-431)
sage: K.class_number()
21
sage: K.hilbert_class_field_defining_polynomial(name='z')
z^21 + z^20 - 13*z^19 - 50*z^18 + 592*z^17 - 2403*z^16 + 5969*z^15 - 10327*z^14 + 13253*z^13 - 12977*z^12 + 9066*z^11 - 2248*z^10 - 5523*z^9 + 11541*z^8 - 13570*z^7 + 11315*z^6 - 6750*z^5 + 2688*z^4 - 577*z^3 + 9*z^2 + 15*z + 1

hilbert_class_polynomial(name='x')¶

Compute the Hilbert class polynomial of this quadratic field.

Right now, this is only implemented for imaginary quadratic fields.

EXAMPLES:

sage: K.<a> = QuadraticField(-3)
sage: K.hilbert_class_polynomial()
x

sage: K.<a> = QuadraticField(-31)
sage: K.hilbert_class_polynomial(name='z')
z^3 + 39491307*z^2 - 58682638134*z + 1566028350940383

is_galois()¶

Return True since all quadratic fields are automatically Galois.

EXAMPLES:

sage: QuadraticField(1234,'d').is_galois()
True

sage.rings.number_field.number_field.NumberField_quadratic_v1(poly, name, canonical_embedding=None)¶

This is used in pickling quadratic fields.

EXAMPLES:

sage: from sage.rings.number_field.number_field import NumberField_quadratic_v1
sage: R.<x> = QQ[]
sage: NumberField_quadratic_v1(x^2 - 2, 'd')
Number Field in d with defining polynomial x^2 - 2

sage.rings.number_field.number_field.QuadraticField(D, names, check=True, embedding=True, latex_name='sqrt')¶

Return a quadratic field obtained by adjoining a square root of $D$ to the rational numbers, where $D$ is not a perfect square.

INPUT:

D - a rational number
names - variable name
check - bool (default: True)
embedding - bool or square root of D in an ambient field (default: True)
latex_name - latex variable name (defalt: sqrt{D})

OUTPUT: A number field defined by a quadratic polynomial. Unless otherwise specified, it has an embedding into $\RR$ or $\CC$ by sending the generator to the positive or upper-half-plane root.

EXAMPLES:

sage: QuadraticField(3, 'a')
Number Field in a with defining polynomial x^2 - 3
sage: K.<theta> = QuadraticField(3); K
Number Field in theta with defining polynomial x^2 - 3
sage: RR(theta)
1.73205080756888
sage: QuadraticField(9, 'a')
...
ValueError: D must not be a perfect square.
sage: QuadraticField(9, 'a', check=False)
Number Field in a with defining polynomial x^2 - 9

Quadratic number fields derive from general number fields.

sage: from sage.rings.number_field.number_field import is_NumberField
sage: type(K)
<class 'sage.rings.number_field.number_field.NumberField_quadratic_with_category'>
sage: is_NumberField(K)
True

Quadratic number fields are cached:

sage: QuadraticField(-11, 'a') is QuadraticField(-11, 'a')
True

By default, quadratic fields come with a nice latex representation:

sage: K.<a> = QuadraticField(-7)
sage: latex(a)
\sqrt{-7}
sage: latex(1/(1+a))
-\frac{1}{8} \sqrt{-7} + \frac{1}{8}
sage: K.latex_variable_name()
'\\sqrt{-7}'

We can provide our own name as well:

sage: K.<a> = QuadraticField(next_prime(10^10), latex_name=r'\sqrt{D}')
sage: 1+a
a + 1
sage: latex(1+a)
\sqrt{D} + 1
sage: latex(QuadraticField(-1, 'a', latex_name=None).gen())
a

TESTS:

sage: QuadraticField(-11, 'a') is QuadraticField(-11, 'a', latex_name='Z')
False
sage: QuadraticField(-11, 'a') is QuadraticField(-11, 'a', latex_name=None)
False

sage.rings.number_field.number_field.gp()¶

Return the unique copy of the gp (PARI) interpreter used for number field computations.

EXAMPLES:

sage: from sage.rings.number_field.number_field import gp
sage: gp()
GP/PARI interpreter

sage.rings.number_field.number_field.is_AbsoluteNumberField(x)¶

Return True if x is an absolute number field.

EXAMPLES:

sage: from sage.rings.number_field.number_field import is_AbsoluteNumberField    
sage: is_AbsoluteNumberField(NumberField(x^2+1,'a'))
True
sage: is_AbsoluteNumberField(NumberField([x^3 + 17, x^2+1],'a'))
False

The rationals are a number field, but they’re not of the absolute number field class.

sage: is_AbsoluteNumberField(QQ)
False

sage.rings.number_field.number_field.is_CyclotomicField(x)¶

Return True if x is a cyclotomic field, i.e., of the special cyclotomic field class. This function does not return True for a number field that just happens to be isomorphic to a cyclotomic field.

EXAMPLES:

sage: from sage.rings.number_field.number_field import is_CyclotomicField
sage: is_CyclotomicField(NumberField(x^2 + 1,'zeta4'))
False
sage: is_CyclotomicField(CyclotomicField(4))
True
sage: is_CyclotomicField(CyclotomicField(1))
True
sage: is_CyclotomicField(QQ)
False
sage: is_CyclotomicField(7)
False

sage.rings.number_field.number_field.is_NumberFieldHomsetCodomain(codomain)¶

Returns whether codomain is a valid codomain for a number field homset. This is used by NumberField._Hom_ to determine whether the created homsets should be a sage.rings.number_field.morphism.NumberFieldHomset.

EXAMPLES:

This currently accepts any parent (CC, RR, ...) in Fields:

sage: from sage.rings.number_field.number_field import is_NumberFieldHomsetCodomain
sage: is_NumberFieldHomsetCodomain(QQ)
True
sage: is_NumberFieldHomsetCodomain(NumberField(x^2 + 1, 'x'))
True
sage: is_NumberFieldHomsetCodomain(ZZ)
False
sage: is_NumberFieldHomsetCodomain(3)
False
sage: is_NumberFieldHomsetCodomain(MatrixSpace(QQ, 2))
False
sage: is_NumberFieldHomsetCodomain(InfinityRing)
False

Question: should, for example, QQ-algebras be accepted as well?

Caveat: Gap objects are not (yet) in Fields, and therefore not accepted as number field homset codomains:

sage: is_NumberFieldHomsetCodomain(gap.Rationals)
False

sage.rings.number_field.number_field.is_QuadraticField(x)¶

Return True if x is of the quadratic number field type.

EXAMPLES:

sage: from sage.rings.number_field.number_field import is_QuadraticField
sage: is_QuadraticField(QuadraticField(5,'a'))
True
sage: is_QuadraticField(NumberField(x^2 - 5, 'b'))
True
sage: is_QuadraticField(NumberField(x^3 - 5, 'b'))
False

A quadratic field specially refers to a number field, not a finite field:

sage: is_QuadraticField(GF(9,'a'))
False

sage.rings.number_field.number_field.is_fundamental_discriminant(D)¶

Return True if the integer $D$ is a fundamental discriminant, i.e., if $D \cong 0,1\pmod{4}$ , and $D\neq 0, 1$ and either (1) $D$ is square free or (2) we have $D\cong 0\pmod{4}$ with $D/4 \cong 2,3\pmod{4}$ and $D/4$ square free. These are exactly the discriminants of quadratic fields.

EXAMPLES:

sage: [D for D in range(-15,15) if is_fundamental_discriminant(D)]
[-15, -11, -8, -7, -4, -3, 5, 8, 12, 13]
sage: [D for D in range(-15,15) if not is_square(D) and QuadraticField(D,'a').disc() == D]
[-15, -11, -8, -7, -4, -3, 5, 8, 12, 13]

sage.rings.number_field.number_field.proof_flag(t)¶

Used for easily determining the correct proof flag to use.

Returns t if t is not None, otherwise returns the system-wide proof-flag for number fields (default: True).

EXAMPLES:

sage: from sage.rings.number_field.number_field import proof_flag
sage: proof_flag(True)
True
sage: proof_flag(False)
False
sage: proof_flag(None)
True
sage: proof_flag("banana")
'banana'

sage.rings.number_field.number_field.put_natural_embedding_first(v)¶

Helper function for embeddings() functions for number fields.

INPUT: a list of embeddings of a number field

OUTPUT: None. The list is altered in-place, so that, if possible, the first embedding has been switched with one of the others, so that if there is an embedding which preserves the generator names then it appears first.

EXAMPLES:

sage: K.<a> = CyclotomicField(7)
sage: embs = K.embeddings(K)
sage: [e(a) for e in embs] # random - there is no natural sort order
[a, a^2, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1]
sage: id = [ e for e in embs if e(a) == a ][0]; id
Ring endomorphism of Cyclotomic Field of order 7 and degree 6
  Defn: a |--> a
sage: permuted_embs = list(embs); permuted_embs.remove(id); permuted_embs.append(id)
sage: [e(a) for e in permuted_embs] # random - but natural map is not first
[a^2, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1, a]
sage: permuted_embs[0] != a
True
sage: from sage.rings.number_field.number_field import put_natural_embedding_first
sage: put_natural_embedding_first(permuted_embs)
sage: [e(a) for e in permuted_embs] # random - but natural map is first
[a, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1, a^2]
sage: permuted_embs[0] == id
True

sage.rings.number_field.number_field.refine_embedding(e, prec=None)¶

Given an embedding from a number field to either $\RR$ or $\CC$ , returns an equivalent embedding with higher precision.

INPUT:

e - an embedding of a number field into either RR or CC (with some precision)
prec - (default None) the desired precision; if None,

current precision is doubled; if Infinity, the equivalent embedding into either QQbar or AA is returned.

EXAMPLES:

sage: from sage.rings.number_field.number_field import refine_embedding
sage: K = CyclotomicField(3)
sage: e10 = K.complex_embedding(10)
sage: e10.codomain().precision()
10
sage: e25 = refine_embedding(e10, prec=25)
sage: e25.codomain().precision()
25

An example where we extend a real embedding into AA:

sage: K.<a> = NumberField(x^3-2)
sage: K.signature()
(1, 1)
sage: e = K.embeddings(RR)[0]; e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Real Field with 53 bits of precision
Defn: a |--> 1.25992104989487
sage: e = refine_embedding(e,Infinity); e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Algebraic Real Field
Defn: a |--> 1.259921049894873?

Now we can obtain arbitrary precision values with no trouble:

sage: RealField(150)(e(a))
1.2599210498948731647672106072782283505702515
sage: _^3
2.0000000000000000000000000000000000000000000
sage: RealField(200)(e(a^2-3*a+7))
4.8076379022835799804500738174376232086807389337953290695624

Complex embeddings can be extended into QQbar:

sage: e = K.embeddings(CC)[0]; e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Complex Field with 53 bits of precision
Defn: a |--> -0.62996052494743... - 1.09112363597172*I
sage: e = refine_embedding(e,Infinity); e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Algebraic Field
Defn: a |--> -0.6299605249474365? - 1.091123635971722?*I
sage: ComplexField(200)(e(a))
-0.62996052494743658238360530363911417528512573235075399004099 - 1.0911236359717214035600726141898088813258733387403009407036*I
sage: e(a)^3
2

Embeddings into lazy fields work:

sage: L = CyclotomicField(7)
sage: x = L.specified_complex_embedding(); x
Generic morphism:                                                                
  From: Cyclotomic Field of order 7 and degree 6                                 
  To:   Complex Lazy Field                                                       
  Defn: zeta7 -> 0.623489801858734? + 0.781831482468030?*I                       
sage: refine_embedding(x, 300)
Ring morphism:
  From: Cyclotomic Field of order 7 and degree 6 
  To:   Complex Field with 300 bits of precision
  Defn: zeta7 |--> 0.623489801858733530525004884004239810632274730896402105365549439096853652456487284575942507 + 0.781831482468029808708444526674057750232334518708687528980634958045091731633936441700868007*I
sage: refine_embedding(x, infinity)
Ring morphism:
  From: Cyclotomic Field of order 7 and degree 6
  To:   Algebraic Field
  Defn: zeta7 |--> 0.6234898018587335? + 0.7818314824680299?*I

Number Fields¶

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