Number Field Ideals¶

AUTHORS:

Steven Sivek (2005-05-16)
William Stein (2007-09-06): vastly improved the doctesting
William Stein and John Cremona (2007-01-28): new class NumberFieldFractionalIdeal now used for all except the 0 ideal

We test that pickling works:

sage: K.<a> = NumberField(x^2 - 5)
sage: I = K.ideal(2/(5+a))
sage: I == loads(dumps(I))
True

class sage.rings.number_field.number_field_ideal.LiftMap(OK, M_OK_map, Q, I)¶: Class to hold data needed by lifting maps from residue fields to number field orders.

class sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal(field, gens, coerce=True)¶

Bases: sage.rings.number_field.number_field_ideal.NumberFieldIdeal

A fractional ideal in a number field.

denominator()¶

Return the denominator ideal of this fractional ideal. Each fractional ideal has a unique expression as $N/D$ where $N$ , $D$ are coprime integral ideals; the denominator is $D$ .

EXAMPLES:

sage: K.<i>=NumberField(x^2+1)
sage: I = K.ideal((3+4*i)/5); I
Fractional ideal (4/5*i + 3/5)
sage: I.denominator()
Fractional ideal (2*i + 1)
sage: I.numerator()
Fractional ideal (-i - 2)
sage: I.numerator().is_integral() and I.denominator().is_integral()
True
sage: I.numerator() + I.denominator() == K.unit_ideal()
True
sage: I.numerator()/I.denominator() == I
True

divides(other)¶

Returns True if this ideal divides other and False otherwise.

EXAMPLES:

sage: K.<a> = CyclotomicField(11); K
Cyclotomic Field of order 11 and degree 10
sage: I = K.factor(31)[0][0]; I
Fractional ideal (-3*a^7 - 4*a^5 - 3*a^4 - 3*a^2 - 3*a - 3)
sage: I.divides(I)
True
sage: I.divides(31)
True
sage: I.divides(29)
False            

element_1_mod(other)¶

Returns an element $r$ in this ideal such that $1-r$ is in other

An error is raised if either ideal is not integral of if they are not coprime.

INPUT:

other – another ideal of the same field, or generators of an ideal.

OUTPUT:

An element $r$ of the ideal self such that $1-r$ is in the ideal other

AUTHOR: Maite Aranes (modified to use PARI’s idealaddtoone by Francis Clarke)

EXAMPLES:

sage: K.<a> = NumberField(x^3-2)
sage: A = K.ideal(a+1); A; A.norm()
Fractional ideal (a + 1)
3
sage: B = K.ideal(a^2-4*a+2); B; B.norm()
Fractional ideal (a^2 - 4*a + 2)
68
sage: r = A.element_1_mod(B); r
a^2 + 5
sage: r in A
True
sage: 1-r in B
True

TESTS:

sage: K.<a> = NumberField(x^3-2)
sage: A = K.ideal(a+1)
sage: B = K.ideal(a^2-4*a+1); B; B.norm()
Fractional ideal (a^2 - 4*a + 1)
99
sage: A.element_1_mod(B)
...
TypeError: Fractional ideal (a + 1), Fractional ideal (a^2 - 4*a + 1) are not coprime ideals

sage: B = K.ideal(1/a); B
Fractional ideal (1/2*a^2)
sage: A.element_1_mod(B)
...
TypeError: Fractional ideal (1/2*a^2) is not an integral ideal

euler_phi()¶

Returns the Euler $\varphi$ -function of this integral ideal.

This is the order of the multiplicative group of the quotient modulo the ideal.

An error is raised if the ideal is not integral.

EXAMPLES:

sage: K.<i>=NumberField(x^2+1)
sage: I = K.ideal(2+i)
sage: [r for r in I.residues() if I.is_coprime(r)]
[-2*i, -i, i, 2*i]
sage: I.euler_phi()
4
sage: J = I^3
sage: J.euler_phi()
100
sage: len([r for r in J.residues() if J.is_coprime(r)])
100
sage: J = K.ideal(3-2*i)
sage: I.is_coprime(J)
True
sage: I.euler_phi()*J.euler_phi() == (I*J).euler_phi()
True
sage: L.<b> = K.extension(x^2 - 7)
sage: L.ideal(3).euler_phi()
64

factor()¶

Factorization of this ideal in terms of prime ideals.

EXAMPLES:

sage: K.<a> = NumberField(x^4 + 23); K
Number Field in a with defining polynomial x^4 + 23
sage: I = K.ideal(19); I
Fractional ideal (19)
sage: F = I.factor(); F
(Fractional ideal (a^2 + 2*a + 2)) * (Fractional ideal (a^2 - 2*a + 2))
sage: type(F)
<class 'sage.structure.factorization.Factorization'>
sage: list(F)
[(Fractional ideal (a^2 + 2*a + 2), 1), (Fractional ideal (a^2 - 2*a + 2), 1)]
sage: F.prod()
Fractional ideal (19)

idealcoprime(J)¶

Returns l such that l*self is coprime to J.

INPUT:

J - another integral ideal of the same field as self, which must also be integral.

OUTPUT:

l - an element such that l*self is coprime to the ideal J

TODO: Extend the implementation to non-integral ideals.

EXAMPLES:

sage: k.<a> = NumberField(x^2 + 23)
sage: A = k.ideal(a+1)
sage: B = k.ideal(3)
sage: A.is_coprime(B)
False
sage: lam = A.idealcoprime(B); lam
-1/6*a + 1/6
sage: (lam*A).is_coprime(B)
True

ALGORITHM: Uses Pari function idealcoprime.

ideallog(x)¶

Returns the discrete logarithm of x with respect to the generators given in the bid structure of the ideal self.

INPUT:

x - a non-zero element of the number field of self, which must have valuation equal to 0 at all prime ideals in the support of the ideal self.

OUTPUT:

l - a list of integers $(x_i)$ such that $0 \leq x_i < d_i$ and $x = \prod_i g_i^{x_i}$ in $(R/I)^*$ , where $I$ = self, $R$ = ring of integers of the field, and $g_i$ are the generators of $(R/I)^*$ , of orders $d_i$ respectively, as given in the bid structure of the ideal self.

EXAMPLES:

sage: k.<a> = NumberField(x^3 - 11)
sage: A = k.ideal(5)
sage: G = A.idealstar(2)
sage: l = A.ideallog(a^2 +3)
sage: r = prod([G.gens()[i]**l[i] for i in range(len(l))])
sage: (a^2 + 3) - r in A
True
sage: A.small_residue(r) # random
a^2 - 2

ALGORITHM: Uses Pari function ideallog

idealstar(flag=1)¶

Returns the finite abelian group $(O_K/I)^*$ , where I is the ideal self of the number field K, and $O_K$ is the ring of integers of K.

INPUT:

flag (int default 1) – when flag =2, it also computes the generators of the group $(O_K/I)^*$ , which takes more time. By default flag =1 (no generators are computed). In both cases the special pari structure bid is computed as well. If flag =0 (deprecated) it computes only the group structure of $(O_K/I)^*$ (with generators) and not the special bid structure.

OUTPUT:

The finite abelian group $(O_K/I)^*$ .

Note

Uses the pari function idealstar. The pari function outputs a special bid structure which is stored in the internal field _bid of the ideal (when flag=1,2). The special structure bid is used in the pari function ideallog to compute discrete logarithms.

EXAMPLES:

sage: k.<a> = NumberField(x^3 - 11)
sage: A = k.ideal(5)
sage: G = A.idealstar(); G
Multiplicative Abelian Group isomorphic to C24 x C4
sage: G.gens()
(f0, f1)
sage: G = A.idealstar(2)
sage: all([G.gens()[i] in k for i in range(G.ngens())])
True

ALGORITHM: Uses Pari function idealstar

invertible_residues(reduce=True)¶

Returns a iterator through a list of invertible residues modulo this integral ideal.

An error is raised if this fractional ideal is not integral.

INPUT:

reduce - bool. If True (default), use small_residue to get small representatives of the residues.

OUTPUT:

An iterator through a list of invertible residues modulo this ideal $I$ , i.e. a list of elements in the ring of integers $R$ representing the elements of $(R/I)^*$ .

ALGORITHM: Use pari’s idealstar to find the group structure and generators of the multiplicative group modulo the ideal.

EXAMPLES:

sage: K.<i>=NumberField(x^2+1)
sage: ires =  K.ideal(2).invertible_residues(); ires  # random address
<generator object at 0xa2feb6c>
sage: list(ires)
[-1, -i]
sage: list(K.ideal(2+i).invertible_residues())
[1, 2, -1, -2]
sage: list(K.ideal(i).residues())
[0]
sage: list(K.ideal(i).invertible_residues())
[1]
sage: I = K.ideal(3+6*i)
sage: units=I.invertible_residues()
sage: len(list(units))==I.euler_phi()
True

sage: K.<a> = NumberField(x^3-10)
sage: I = K.ideal(a-1)
sage: len(list(I.invertible_residues())) == I.euler_phi()
True

sage: K.<z> = CyclotomicField(10)
sage: len(list(K.primes_above(3)[0].invertible_residues())) 
80

AUTHOR: John Cremona

invertible_residues_mod(subgp_gens=[], reduce=True)¶

Returns a iterator through a list of representatives for the invertible residues modulo this integral ideal, modulo the subgroup generated by the elements in the list subgp_gens.

INPUT:

subgp_gens - either None or a list of elements of the number field of self. These need not be integral, but should be coprime to the ideal self. If the list is empty or None, the function returns an iterator through a list of representatives for the invertible residues modulo the integral ideal self.
reduce - bool. If True (default), use small_residues to get small representatives of the residues.

Note

See also invertible_residues() for a simpler version without the subgroup.

OUTPUT:

An iterator through a list of representatives for the invertible residues modulo self and modulo the group generated by subgp_gens, i.e. a list of elements in the ring of integers $R$ representing the elements of $(R/I)^*/U$ , where $I$ is this ideal and $U$ is the subgroup of $(R/I)^*$ generated by subgp_gens.

EXAMPLES:

sage: k.<a> = NumberField(x^2 +23)
sage: I = k.ideal(a)
sage: list(I.invertible_residues_mod([-1]))
[1, 5, 2, 10, 4, -3, 8, -6, -7, 11, 9]
sage: list(I.invertible_residues_mod([1/2]))
[1, 5]
sage: list(I.invertible_residues_mod([23]))
...
TypeError: the element must be invertible mod the ideal

sage: K.<a> = NumberField(x^3-10)
sage: I = K.ideal(a-1)
sage: len(list(I.invertible_residues_mod([]))) == I.euler_phi()
True

sage: I = K.ideal(1)
sage: list(I.invertible_residues_mod([]))
[1]

sage: K.<z> = CyclotomicField(10)
sage: len(list(K.primes_above(3)[0].invertible_residues_mod([])))
80

AUTHOR: Maite Aranes.

is_coprime(other)¶

Returns True if this ideal is coprime to the other, else False.

INPUT:

other – another ideal of the same field, or generators of an ideal.

OUTPUT:

True if self and other are coprime, else False.

Note

This function works for fractional ideals as well as integral ideals.

AUTHOR: John Cremona

EXAMPLES:

sage: K.<i>=NumberField(x^2+1)
sage: I = K.ideal(2+i)
sage: J = K.ideal(2-i)
sage: I.is_coprime(J)
True
sage: (I^-1).is_coprime(J^3)
True
sage: I.is_coprime(5)
False
sage: I.is_coprime(6+i)
True

# See trac \# 4536:
sage: E.<a> = NumberField(x^5 + 7*x^4 + 18*x^2 + x - 3)
sage: OE = E.ring_of_integers()
sage: i,j,k = [u[0] for u in factor(3*OE)]
sage: (i/j).is_coprime(j/k)
False
sage: (j/k).is_coprime(j/k)
False

sage: F.<a, b> = NumberField([x^2 - 2, x^2 - 3])
sage: F.ideal(3 - a*b).is_coprime(F.ideal(3))
False

is_maximal()¶

Return True if this ideal is maximal. This is equivalent to self being prime, since it is nonzero.

EXAMPLES:

sage: K.<a> = NumberField(x^3 + 3); K
Number Field in a with defining polynomial x^3 + 3
sage: K.ideal(5).is_maximal()
False
sage: K.ideal(7).is_maximal()
True        

is_trivial(proof=None)¶

Returns True if this is a trivial ideal.

EXAMPLES:

sage: F.<a> = QuadraticField(-5)
sage: I = F.ideal(3)
sage: I.is_trivial()
False
sage: J = F.ideal(5)
sage: J.is_trivial()
False
sage: (I+J).is_trivial()
True

numerator()¶

Return the numerator ideal of this fractional ideal.

Each fractional ideal has a unique expression as $N/D$ where $N$ , $D$ are coprime integral ideals. The numerator is $N$ .

EXAMPLES:

sage: K.<i>=NumberField(x^2+1)
sage: I = K.ideal((3+4*i)/5); I
Fractional ideal (4/5*i + 3/5)
sage: I.denominator()
Fractional ideal (2*i + 1)
sage: I.numerator()
Fractional ideal (-i - 2)
sage: I.numerator().is_integral() and I.denominator().is_integral()
True
sage: I.numerator() + I.denominator() == K.unit_ideal()
True
sage: I.numerator()/I.denominator() == I
True

prime_factors()¶

Return a list of the prime ideal factors of self

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

EXAMPLES:

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

prime_to_idealM_part(M)¶

Version for integral ideals of the prime_to_m_part function over $\ZZ$ . Returns the largest divisor of self that is coprime to the ideal M.

INPUT:

M – an integral ideal of the same field, or generators of an ideal

OUTPUT:

An ideal which is the largest divisor of self that is coprime to $M$ .

AUTHOR: Maite Aranes

EXAMPLES:

sage: k.<a> = NumberField(x^2 + 23)
sage: I = k.ideal(a+1)
sage: M = k.ideal(2, 1/2*a - 1/2)
sage: J = I.prime_to_idealM_part(M); J
Fractional ideal (12, 1/2*a + 13/2)
sage: J.is_coprime(M)
True

sage: J = I.prime_to_idealM_part(2); J
Fractional ideal (3, -1/2*a - 1/2)
sage: J.is_coprime(M)
True

ramification_index()¶

Return the ramification index of this fractional ideal, assuming it is prime. Otherwise, raise a ValueError.

The ramification index is the power of this prime appearing in the factorization of the prime in $\ZZ$ that this prime lies over.

EXAMPLES:

sage: K.<a> = NumberField(x^2 + 2); K
Number Field in a with defining polynomial x^2 + 2
sage: f = K.factor(2); f
(Fractional ideal (-a))^2
sage: f[0][0].ramification_index()
2
sage: K.ideal(13).ramification_index()
1
sage: K.ideal(17).ramification_index()
...
ValueError: the fractional ideal (= Fractional ideal (17)) is not prime

reduce(f)¶

Return the canonical reduction of the element of $f$ modulo the ideal $I$ (=self). This is an element of $R$ (the ring of integers of the number field) that is equivalent modulo $I$ to $f$ .

An error is raised if this fractional ideal is not integral or the element $f$ is not integral.

INPUT:

f - an integral element of the number field

OUTPUT:

An integral element $g$ , such that $f - g$ belongs to the ideal self and such that $g$ is a canonical reduced representative of the coset $f + I$ ( $I$ =self) as described in the residues function, namely an integral element with coordinates $(r_0, \dots,r_{n-1})$ , where:

$r_i$ is reduced modulo $d_i$
$d_i = b_i[i]$ , with ${b_0, b_1, \dots, b_n}$ HNF basis of the ideal self.

Note

The reduced element $g$ is not necessarily small. To get a small $g$ use the method small_residue.

EXAMPLES:

sage: k.<a> = NumberField(x^3 + 11)
sage: I = k.ideal(5, a^2 - a + 1)
sage: c = 4*a + 9
sage: I.reduce(c)
a^2 - 2*a
sage: c - I.reduce(c) in I
True

The reduced element is in the list of canonical representatives returned by the residues method:

sage: I.reduce(c) in list(I.residues())
True

The reduced element does not necessarily have smaller norm (use small_residue for that)

sage: c.norm()
25
sage: (I.reduce(c)).norm()
209
sage: (I.small_residue(c)).norm()
10

Sometimes the canonical reduced representative of $1$ won’t be $1$ (it depends on the choice of basis for the ring of integers):

sage: k.<a> = NumberField(x^2 + 23)
sage: I = k.ideal(3)
sage: I.reduce(3*a + 1)
-3/2*a - 1/2
sage: k.ring_of_integers().basis()
[1/2*a + 1/2, a]

AUTHOR: Maite Aranes.

residue_class_degree()¶

Return the residue class degree of this fractional ideal, assuming it is prime. Otherwise, raise a ValueError.

The residue class degree of a prime ideal $I$ is the degree of the extension $O_K/I$ of its prime subfield.

EXAMPLES:

sage: K.<a> = NumberField(x^5 + 2); K
Number Field in a with defining polynomial x^5 + 2
sage: f = K.factor(19); f
(Fractional ideal (a^2 + a - 3)) * (Fractional ideal (-2*a^4 - a^2 + 2*a - 1)) * (Fractional ideal (a^2 + a - 1))
sage: [i.residue_class_degree() for i, _ in f]   
[2, 2, 1]        

residue_field(names=None)¶

Return the residue class field of this fractional ideal, which must be prime.

EXAMPLES:

sage: K.<a> = NumberField(x^3-7)
sage: P = K.ideal(29).factor()[0][0]
sage: P.residue_field()
Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)
sage: P.residue_field('z')
Residue field in z of Fractional ideal (2*a^2 + 3*a - 10)

Another example:

sage: K.<a> = NumberField(x^3-7)
sage: P = K.ideal(389).factor()[0][0]; P
Fractional ideal (389, a^2 - 44*a - 9)
sage: P.residue_class_degree()
2
sage: P.residue_field()
Residue field in abar of Fractional ideal (389, a^2 - 44*a - 9)
sage: P.residue_field('z')
Residue field in z of Fractional ideal (389, a^2 - 44*a - 9)
sage: FF.<w> = P.residue_field()
sage: FF
Residue field in w of Fractional ideal (389, a^2 - 44*a - 9)
sage: FF((a+1)^390)
36
sage: FF(a)
w

An example of reduction maps to the residue field: these are defined on the whole valuation ring, i.e. the subring of the number field consisting of elements with non-negative valuation. This shows that the issue raised in trac #1951 has been fixed:

sage: K.<i> = NumberField(x^2 + 1)
sage: P1, P2 = [g[0] for g in K.factor(5)]; (P1,P2)
(Fractional ideal (-i - 2), Fractional ideal (2*i + 1))
sage: a = 1/(1+2*i)
sage: F1, F2 = [g.residue_field() for g in [P1,P2]]; (F1,F2) 
(Residue field of Fractional ideal (-i - 2),
Residue field of Fractional ideal (2*i + 1))
sage: a.valuation(P1)
0
sage: F1(i/7)
4
sage: F1(a)
3
sage: a.valuation(P2)
-1
sage: F2(a)
Traceback (most recent call last):
ZeroDivisionError: Cannot reduce field element -2/5*i + 1/5 modulo Fractional ideal (2*i + 1): it has negative valuation        

An example with a relative number field:

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: R = p.residue_field(); R
Residue field in abar of Fractional ideal ((-1/2*b - 1/2)*a + 1/2*b - 1/2)
sage: R.cardinality()
9
sage: R(17)
2
sage: R((a + b)/17)
abar
sage: R(1/b)
2*abar

residues()¶

Returns a iterator through a complete list of residues modulo this integral ideal.

An error is raised if this fractional ideal is not integral.

OUTPUT:

An iterator through a complete list of residues modulo the integral ideal self. This list is the set of canonical reduced representatives given by all integral elements with coordinates $(r_0, \dots,r_{n-1})$ , where:

$r_i$ is reduced modulo $d_i$
$d_i = b_i[i]$ , with ${b_0, b_1, \dots, b_n}$ HNF basis of the ideal.

AUTHOR: John Cremona (modified by Maite Aranes)

EXAMPLES:

sage: K.<i>=NumberField(x^2+1)
sage: res =  K.ideal(2).residues(); res  # random address
xmrange_iter([[0, 1], [0, 1]], <function <lambda> at 0xa252144>)
sage: list(res)
[0, i, 1, i + 1]
sage: list(K.ideal(2+i).residues())
[-2*i, -i, 0, i, 2*i]
sage: list(K.ideal(i).residues())
[0]
sage: I = K.ideal(3+6*i)
sage: reps=I.residues()
sage: len(list(reps)) == I.norm()
True
sage: all([r==s or not (r-s) in I for r in reps for s in reps])
True

sage: K.<a> = NumberField(x^3-10)
sage: I = K.ideal(a-1)
sage: len(list(I.residues())) == I.norm()
True

sage: K.<z> = CyclotomicField(11)
sage: len(list(K.primes_above(3)[0].residues())) == 3**5
True

small_residue(f)¶

Given an element $f$ of the ambient number field, returns an element $g$ such that $f - g$ belongs to the ideal self (which must be integral), and $g$ is small.

Note

The reduced representative returned is not uniquely determined.

ALGORITHM: Uses Pari function nfeltreduce.

EXAMPLES:

sage: k.<a> = NumberField(x^2 + 5)
sage: I = k.ideal(a)
sage: I.small_residue(14)
-1

sage: K.<a> = NumberField(x^5 + 7*x^4 + 18*x^2 + x - 3)
sage: I = K.ideal(5)
sage: I.small_residue(a^2 -13)
a^2 + 5*a - 3

class sage.rings.number_field.number_field_ideal.NumberFieldIdeal(field, gens, coerce=True)¶

Bases: sage.rings.ideal.Ideal_generic

An ideal of a number field.

absolute_norm()¶

A synonym for norm.

EXAMPLES:

sage: K.<i> = NumberField(x^2 + 1)
sage: K.ideal(1 + 2*i).absolute_norm()
5

absolute_ramification_index()¶

A synonym for ramification_index.

EXAMPLES:

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

artin_symbol()¶

Return the Artin symbol $( K / \QQ, P)$ , where $K$ is the number field of $P$ =self. This is the unique element $s$ of the decomposition group of $P$ such that $s(x) = x^p \pmod{P}$ where $p$ is the residue characteristic of $P$ . (Here $P$ (self) should be prime and unramified.)

See the artin_symbol method of the GaloisGroup_v2 class for further documentation and examples.

EXAMPLE:

sage: QuadraticField(-23, 'w').primes_above(7)[0].artin_symbol()
(1,2)

basis()¶

Return an immutable sequence of elements of this ideal (note: their parent is the number field) that form a basis for this ideal viewed as a $\ZZ$ -module.

OUTPUT:: basis – an immutable sequence.

EXAMPLES:

sage: K.<z> = CyclotomicField(7)
sage: I = K.factor(11)[0][0]
sage: I.basis()           # warning -- choice of basis can be somewhat random
[11, 11*z, 11*z^2, z^3 + 5*z^2 + 4*z + 10, z^4 + z^2 + z + 5, z^5 + z^4 + z^3 + 2*z^2 + 6*z + 5]

An example of a non-integral ideal.:

sage: J = 1/I
sage: J          # warning -- choice of generators can be somewhat random 
Fractional ideal (2/11*z^5 + 2/11*z^4 + 3/11*z^3 + 2/11)
sage: J.basis()           # warning -- choice of basis can be somewhat random
[1, z, z^2, 1/11*z^3 + 7/11*z^2 + 6/11*z + 10/11, 1/11*z^4 + 1/11*z^2 + 1/11*z + 7/11, 1/11*z^5 + 1/11*z^4 + 1/11*z^3 + 2/11*z^2 + 8/11*z + 7/11]

coordinates(x)¶

Returns the coordinate vector of $x$ with respect to this ideal.

INPUT:: x – an element of the number field (or ring of integers) of this ideal.
OUTPUT:: A vector of length $n$ (the degree of the field) giving the coordinates of $x$ with respect to the integral basis of the ideal. In general this will be a vector of rationals; it will consist of integers if and only if $x$ is in the ideal.

AUTHOR: John Cremona 2008-10-31

ALGORITHM:

Uses linear algebra. The change-of-basis matrix is cached. Provides simpler implementations for _contains_(), is_integral() and smallest_integer().

EXAMPLES:

sage: K.<i> = QuadraticField(-1)
sage: I = K.ideal(7+3*i)
sage: Ibasis = I.integral_basis(); Ibasis
[58, i + 41]
sage: a = 23-14*i
sage: acoords = I.coordinates(a); acoords
(597/58, -14)
sage: sum([Ibasis[j]*acoords[j] for j in range(2)]) == a
True
sage: b = 123+456*i
sage: bcoords = I.coordinates(b); bcoords
(-18573/58, 456)
sage: sum([Ibasis[j]*bcoords[j] for j in range(2)]) == b
True

decomposition_group()¶

Return the decomposition group of self, as a subset of the automorphism group of the number field of self. Raises an error if the field isn’t Galois. See the decomposition_group method of the GaloisGroup_v2 class for further examples and doctests.

EXAMPLE:

sage: QuadraticField(-23, 'w').primes_above(7)[0].decomposition_group()
Galois group of Number Field in w with defining polynomial x^2 + 23

free_module()¶

Return the free $\ZZ$ -module contained in the vector space associated to the ambient number field, that corresponds to this ideal.

EXAMPLES:

sage: K.<z> = CyclotomicField(7)
sage: I = K.factor(11)[0][0]; I
Fractional ideal (-2*z^4 - 2*z^2 - 2*z + 1)
sage: A = I.free_module()
sage: A              # warning -- choice of basis can be somewhat random
Free module of degree 6 and rank 6 over Integer Ring
User basis matrix:
[11  0  0  0  0  0]
[ 0 11  0  0  0  0]
[ 0  0 11  0  0  0]
[10  4  5  1  0  0]
[ 5  1  1  0  1  0]
[ 5  6  2  1  1  1]

However, the actual $\ZZ$ -module is not at all random:

sage: A.basis_matrix().change_ring(ZZ).echelon_form()
[ 1  0  0  5  1  1]
[ 0  1  0  1  1  7]
[ 0  0  1  7  6 10]
[ 0  0  0 11  0  0]
[ 0  0  0  0 11  0]
[ 0  0  0  0  0 11]

The ideal doesn’t have to be integral:

sage: J = I^(-1)
sage: B = J.free_module()
sage: B.echelonized_basis_matrix()
[ 1/11     0     0  7/11  1/11  1/11]
[    0  1/11     0  1/11  1/11  5/11]
[    0     0  1/11  5/11  4/11 10/11]
[    0     0     0     1     0     0]
[    0     0     0     0     1     0]
[    0     0     0     0     0     1]

This also works for relative extensions:

sage: K.<a,b> = NumberField([x^2 + 1, x^2 + 2])
sage: I = K.fractional_ideal(4)
sage: I.free_module()
Free module of degree 4 and rank 4 over Integer Ring
User basis matrix:
[  4   0   0   0]
[ -3   7  -1   1]
[  3   7   1   1]
[  0 -10   0  -2]
sage: J = I^(-1); J.free_module()
Free module of degree 4 and rank 4 over Integer Ring
User basis matrix:
[  1/4     0     0     0]
[-3/16  7/16 -1/16  1/16]
[ 3/16  7/16  1/16  1/16]
[    0  -5/8     0  -1/8]

An example of intersecting ideals by intersecting free modules.:

sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
sage: I = K.factor(2)
sage: p1 = I[0][0]; p2 = I[1][0]
sage: N = p1.free_module().intersection(p2.free_module()); N
Free module of degree 3 and rank 3 over Integer Ring
Echelon basis matrix:
[  1 1/2 1/2]
[  0   1   1]
[  0   0   2]
sage: N.index_in(p1.free_module()).abs()
2

TESTS:

Sage can find the free module associated to quite large ideals quickly (see trac #4627):

sage: y = polygen(ZZ)
sage: M.<a> = NumberField(y^20 - 2*y^19 + 10*y^17 - 15*y^16 + 40*y^14 - 64*y^13 + 46*y^12 + 8*y^11 - 32*y^10 + 8*y^9 + 46*y^8 - 64*y^7 + 40*y^6 - 15*y^4 + 10*y^3 - 2*y + 1)
sage: M.ideal(prod(prime_range(6000, 6200))).free_module()
Free module of degree 20 and rank 20 over Integer Ring
User basis matrix:
20 x 20 dense matrix over Rational Field

gens_reduced(proof=None)¶

Express this ideal in terms of at most two generators, and one if possible.

Note that if the ideal is not principal, then this uses PARI’s idealtwoelt function, which takes exponential time, the first time it is called for each ideal. Also, this indirectly uses bnfisprincipal, so set proof=True if you want to prove correctness (which is the default).

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: K.<i> = NumberField(x^2+1, 'i')
sage: J = K.ideal([i+1, 2])
sage: J.gens()
(i + 1, 2)
sage: J.gens_reduced()
(i + 1,)

TESTS:

sage: all(j.parent() is K for j in J.gens())
True
sage: all(j.parent() is K for j in J.gens_reduced())
True

sage: K.<a> = NumberField(x^4 + 10*x^2 + 20)
sage: J = K.prime_above(5)
sage: J.is_principal()
False
sage: J.gens_reduced()
(5, a)
sage: all(j.parent() is K for j in J.gens())
True
sage: all(j.parent() is K for j in J.gens_reduced())
True

inertia_group()¶

Return the inertia group of self, i.e. the set of elements s of the Galois group of the number field of self (which we assume is Galois) such that s acts trivially modulo self. This is the same as the 0th ramification group of self. See the inertia_group method of the GaloisGroup_v2 class for further examples and doctests.

EXAMPLE:

sage: QuadraticField(-23, 'w').primes_above(23)[0].inertia_group()
Galois group of Number Field in w with defining polynomial x^2 + 23

integral_basis()¶

Return a list of generators for this ideal as a $\ZZ$ -module.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)        
sage: K.<i> = NumberField(x^2 + 1)
sage: J = K.ideal(i+1)
sage: J.integral_basis()
[2, i + 1]

integral_split()¶

Return a tuple $(I, d)$ , where $I$ is an integral ideal, and $d$ is the smallest positive integer such that this ideal is equal to $I/d$ .

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^2-5)
sage: I = K.ideal(2/(5+a))
sage: I.is_integral()
False
sage: J,d = I.integral_split()
sage: J
Fractional ideal (-1/2*a + 5/2)
sage: J.is_integral()
True
sage: d
5
sage: I == J/d
True

intersection(other)¶

Return the intersection of self and other.

EXAMPLE:

sage: K.<a> = QuadraticField(-11)
sage: p = K.ideal((a + 1)/2); q = K.ideal((a + 3)/2)
sage: p.intersection(q) == q.intersection(p) == K.ideal(a-2)
True

An example with non-principal ideals:

sage: L.<a> = NumberField(x^3 - 7)
sage: p = L.ideal(a^2 + a + 1, 2)
sage: q = L.ideal(a+1)
sage: p.intersection(q) == L.ideal(8, 2*a + 2)
True

A relative example:

sage: L.<a,b> = NumberField([x^2 + 11, x^2 - 5])
sage: A = L.ideal([15, (-3/2*b + 7/2)*a - 8])
sage: B = L.ideal([6, (-1/2*b + 1)*a - b - 5/2])
sage: A.intersection(B) == L.ideal(-1/2*a - 3/2*b - 1)
True

is_integral()¶

Return True if this ideal is integral.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)        
sage: K.<a> = NumberField(x^5-x+1)
sage: K.ideal(a).is_integral()
True
sage: (K.ideal(1) / (3*a+1)).is_integral()
False

is_maximal()¶

Return True if this ideal is maximal. This is equivalent to self being prime and nonzero.

EXAMPLES:

sage: K.<a> = NumberField(x^3 + 3); K
Number Field in a with defining polynomial x^3 + 3
sage: K.ideal(5).is_maximal()
False
sage: K.ideal(7).is_maximal()
True        

is_prime()¶

Return True if this ideal is prime.

EXAMPLES:

sage: K.<a> = NumberField(x^2 - 17); K
Number Field in a with defining polynomial x^2 - 17
sage: K.ideal(5).is_prime()
True
sage: K.ideal(13).is_prime()
False
sage: K.ideal(17).is_prime()
False        

is_principal(proof=None)¶

Return True if this ideal is principal.

Since it uses the PARI method bnfisprincipal, specify proof=True (this is the default setting) to prove the correctness of the output.

EXAMPLES:

sage: K = QuadraticField(-119,’a’) sage: P = K.factor(2)[1][0] sage: P.is_principal() False sage: I = P^5 sage: I.is_principal() True sage: I # random Fractional ideal (-1/2*a + 3/2) sage: P = K.ideal([2]).factor()[1][0] sage: I = P^5 sage: I.is_principal() True

is_zero()¶

Return True iff self is the zero ideal

EXAMPLES:

sage: K.<a> = NumberField(x^2 + 2); K
Number Field in a with defining polynomial x^2 + 2
sage: K.ideal(3).is_zero()
False
sage: I=K.ideal(0); I.is_zero()
True
sage: I
Ideal (0) of Number Field in a with defining polynomial x^2 + 2

(0 is a NumberFieldIdeal, not a NumberFieldFractionIdeal)

norm()¶

Return the norm of this fractional ideal as a rational number.

EXAMPLES:

sage: K.<a> = NumberField(x^4 + 23); K
Number Field in a with defining polynomial x^4 + 23
sage: I = K.ideal(19); I
Fractional ideal (19)
sage: factor(I.norm())
19^4
sage: F = I.factor()
sage: F[0][0].norm().factor()
19^2        

number_field()¶

Return the number field that this is a fractional ideal in.

EXAMPLES:

sage: K.<a> = NumberField(x^2 + 2); K
Number Field in a with defining polynomial x^2 + 2
sage: K.ideal(3).number_field()
Number Field in a with defining polynomial x^2 + 2
sage: K.ideal(0).number_field() # not tested (not implemented)
Number Field in a with defining polynomial x^2 + 2        

pari_hnf()¶

Return PARI’s representation of this ideal in Hermite normal form.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^3 - 2)
sage: I = K.ideal(2/(5+a))
sage: I.pari_hnf()
[2, 0, 50/127; 0, 2, 244/127; 0, 0, 2/127]

ramification_group(v)¶

Return the $v$ ‘th ramification group of self, i.e. the set of elements $s$ of the Galois group of the number field of self (which we assume is Galois) such that $s$ acts trivially modulo the $(v+1)$ ‘st power of self. See the ramification_group method of the GaloisGroup class for further examples and doctests.

EXAMPLE:

sage: QuadraticField(-23, 'w').primes_above(23)[0].ramification_group(0)
Galois group of Number Field in w with defining polynomial x^2 + 23
sage: QuadraticField(-23, 'w').primes_above(23)[0].ramification_group(1)
Subgroup [()] of Galois group of Number Field in w with defining polynomial x^2 + 23

random_element(*args, **kwds)¶

Return a random element of this order.

INPUT:

*args, *kwds - Parameters passed to the random integer function. See the documentation of ZZ.random_element() for details.

OUTPUT:

A random element of this fractional ideal, computed as a random $\ZZ$ -linear combination of the basis.

EXAMPLES:

sage: K.<a> = NumberField(x^3 + 2)
sage: I = K.ideal(1-a)
sage: I.random_element() # random output
-a^2 - a - 19
sage: I.random_element(distribution="uniform") # random output
a^2 - 2*a - 8
sage: I.random_element(-30,30) # random output
-7*a^2 - 17*a - 75
sage: I.random_element(-100, 200).is_integral()
True
sage: I.random_element(-30,30).parent() is K
True

A relative example:

sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 1000*x + 1])
sage: I = K.ideal(1-a)
sage: I.random_element() # random output
17/500002*a^3 + 737253/250001*a^2 - 1494505893/500002*a + 752473260/250001
sage: I.random_element().is_integral()
True
sage: I.random_element(-100, 200).parent() is K
True

reduce_equiv()¶

Return a small ideal that is equivalent to self in the group of fractional ideals modulo principal ideals. Very often (but not always) if self is principal then this function returns the unit ideal.

ALGORITHM: Calls pari’s idealred function.

EXAMPLES:

sage: K.<w> = NumberField(x^2 + 23)
sage: I = ideal(w*23^5); I
Fractional ideal (6436343*w)
sage: I.reduce_equiv()
Fractional ideal (1)
sage: I = K.class_group().0.ideal()^10; I
Fractional ideal (1024, 1/2*w + 979/2)
sage: I.reduce_equiv()
Fractional ideal (2, 1/2*w - 1/2)

relative_norm()¶

A synonym for norm.

EXAMPLES:

sage: K.<i> = NumberField(x^2 + 1)
sage: K.ideal(1 + 2*i).relative_norm()
5

relative_ramification_index()¶

A synonym for ramification_index.

EXAMPLES:

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

smallest_integer()¶

Return the smallest non-negative integer in $I \cap \ZZ$ , where $I$ is this ideal. If $I = 0$ , returns 0.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^2+6)
sage: I = K.ideal([4,a])/7
sage: I.smallest_integer()
2

sage: K.<i> = QuadraticField(-1)
sage: P1, P2 = [P for P,e in K.factor(13)]
sage: all([(P1^i*P2^j).smallest_integer() == 13^max(i,j,0) for i in range(-3,3) for j in range(-3,3)])
True
sage: I = K.ideal(0)
sage: I.smallest_integer()
0

# See trac\# 4392:
sage: K.<a>=QuadraticField(-5)
sage: I=K.ideal(7)
sage: I.smallest_integer()
7

sage: K.<z> = CyclotomicField(13)
sage: a = K([-8, -4, -4, -6, 3, -4, 8, 0, 7, 4, 1, 2])
sage: I = K.ideal(a)
sage: I.smallest_integer()
146196692151
sage: I.norm()
1315770229359
sage: I.norm() / I.smallest_integer()
9

valuation(p)¶

Return the valuation of self at p.

INPUT:

p – a prime ideal $\mathfrak{p}$ of this number field.

OUTPUT:

(integer) The valuation of this fractional ideal at the prime $\mathfrak{p}$ . If $\mathfrak{p}$ is not prime, raise a ValueError.

EXAMPLES:

sage: K.<a> = NumberField(x^5 + 2); K
Number Field in a with defining polynomial x^5 + 2
sage: i = K.ideal(38); i
Fractional ideal (38)
sage: i.valuation(K.factor(19)[0][0])
1
sage: i.valuation(K.factor(2)[0][0])
5
sage: i.valuation(K.factor(3)[0][0])
0
sage: i.valuation(0)
...
ValueError: p (= 0) must be nonzero

class sage.rings.number_field.number_field_ideal.QuotientMap(K, M_OK_change, Q, I)¶: Class to hold data needed by quotient maps from number field orders to residue fields. These are only partial maps: the exact domain is the appropriate valuation ring. For examples, see residue_field().

sage.rings.number_field.number_field_ideal.basis_to_module(B, K)¶

Given a basis $B$ of elements for a $\ZZ$ -submodule of a number field $K$ , return the corresponding $\ZZ$ -submodule.

EXAMPLES:

sage: K.<w> = NumberField(x^4 + 1)
sage: from sage.rings.number_field.number_field_ideal import basis_to_module
sage: basis_to_module([K.0, K.0^2 + 3], K)
Free module of degree 4 and rank 2 over Integer Ring
User basis matrix:
[0 1 0 0]
[3 0 1 0]

sage.rings.number_field.number_field_ideal.convert_from_idealprimedec_form(field, ideal)¶

Used internally in the number field ideal implementation for converting from the form output by the pari function idealprimedec to a Sage ideal.

INPUT:

field - a number field
ideal - a pari ideal, as output by the idealprimedec function

EXAMPLE:

sage: from sage.rings.number_field.number_field_ideal import convert_from_idealprimedec_form
sage: K.<a> = NumberField(x^2 + 3)
sage: K_bnf = gp(K.pari_bnf())
sage: ideal = K_bnf.idealprimedec(3)[1]
sage: convert_from_idealprimedec_form(K, ideal)
Fractional ideal (1/2*a - 3/2)
sage: K.factor(3)
(Fractional ideal (1/2*a - 3/2))^2

sage.rings.number_field.number_field_ideal.convert_from_zk_basis(field, hnf)¶

Used internally in the number field ideal implementation for converting from the order basis to the number field basis.

INPUT:

field - a number field
hnf - a pari HNF matrix, output by the pari_hnf() function.

EXAMPLES:

sage: from sage.rings.number_field.number_field_ideal import convert_from_zk_basis
sage: k.<a> = NumberField(x^2 + 23)
sage: I = k.factor(3)[0][0]; I
Fractional ideal (3, -1/2*a + 1/2)
sage: hnf = I.pari_hnf(); hnf
[3, 0; 0, 1]
sage: convert_from_zk_basis(k, hnf)
[3, 1/2*x - 1/2]

sage.rings.number_field.number_field_ideal.convert_to_idealprimedec_form(field, ideal)¶

Used internally in the number field ideal implementation for converting to the form output by the pari function idealprimedec from a Sage ideal.

INPUT:

field - a number field
ideal - a prime ideal

NOTE:

The algorithm implemented right now is not optimal, but works. It should eventually be replaced with something better.

EXAMPLE:

sage: from sage.rings.number_field.number_field_ideal import convert_to_idealprimedec_form
sage: K.<a> = NumberField(x^2 + 3)
sage: P = K.ideal(a/2-3/2)
sage: convert_to_idealprimedec_form(K, P)
[3, [1, 2]~, 2, 1, [1, -1]~]

sage.rings.number_field.number_field_ideal.is_NumberFieldFractionalIdeal(x)¶

Return True if x is a fractional ideal of a number field.

EXAMPLES:

sage: from sage.rings.number_field.number_field_ideal import is_NumberFieldFractionalIdeal
sage: is_NumberFieldFractionalIdeal(2/3)
False
sage: is_NumberFieldFractionalIdeal(ideal(5))
False
sage: k.<a> = NumberField(x^2 + 2)
sage: I = k.ideal([a + 1]); I
Fractional ideal (a + 1)
sage: is_NumberFieldFractionalIdeal(I)
True
sage: Z = k.ideal(0); Z
Ideal (0) of Number Field in a with defining polynomial x^2 + 2
sage: is_NumberFieldFractionalIdeal(Z)
False

sage.rings.number_field.number_field_ideal.is_NumberFieldIdeal(x)¶

Return True if x is an ideal of a number field.

EXAMPLES:

sage: from sage.rings.number_field.number_field_ideal import is_NumberFieldIdeal
sage: is_NumberFieldIdeal(2/3)
False
sage: is_NumberFieldIdeal(ideal(5))
False
sage: k.<a> = NumberField(x^2 + 2)
sage: I = k.ideal([a + 1]); I
Fractional ideal (a + 1)
sage: is_NumberFieldIdeal(I)
True
sage: Z = k.ideal(0); Z
Ideal (0) of Number Field in a with defining polynomial x^2 + 2
sage: is_NumberFieldIdeal(Z)
True

sage.rings.number_field.number_field_ideal.quotient_char_p(I, p)¶

Given an integral ideal $I$ that contains a prime number $p$ , compute a vector space $V = (O_K \mod p) / (I \mod p)$ , along with a homomorphism $O_K \to V$ and a section $V \to O_K$ .

EXAMPLES:

sage: from sage.rings.number_field.number_field_ideal import quotient_char_p

sage: K.<i> = NumberField(x^2 + 1); O = K.maximal_order(); I = K.fractional_ideal(15)
sage: quotient_char_p(I, 5)[0]
Vector space quotient V/W of dimension 2 over Finite Field of size 5 where
V: Vector space of dimension 2 over Finite Field of size 5
W: Vector space of degree 2 and dimension 0 over Finite Field of size 5
Basis matrix:
[]
sage: quotient_char_p(I, 3)[0]
Vector space quotient V/W of dimension 2 over Finite Field of size 3 where
V: Vector space of dimension 2 over Finite Field of size 3
W: Vector space of degree 2 and dimension 0 over Finite Field of size 3
Basis matrix:
[]

sage: I = K.factor(13)[0][0]; I
Fractional ideal (-3*i - 2)
sage: I.residue_class_degree()
1
sage: quotient_char_p(I, 13)[0]
Vector space quotient V/W of dimension 1 over Finite Field of size 13 where
V: Vector space of dimension 2 over Finite Field of size 13
W: Vector space of degree 2 and dimension 1 over Finite Field of size 13
Basis matrix:
[1 8]

Number Field Ideals¶

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