p-Adic Generic Element.¶

Elements of p-Adic Rings and Fields

AUTHOR:

David Roe
Genya Zaytman: documentation
David Harvey: doctests

sage.rings.padics.padic_generic_element.clear_mpz_globals()¶

sage.rings.padics.padic_generic_element.gmp_randrange(n1, n2)¶

sage.rings.padics.padic_generic_element.init_mpz_globals()¶

class sage.rings.padics.padic_generic_element.pAdicGenericElement¶

Bases: sage.rings.padics.local_generic_element.LocalGenericElement

abs(prec=None)¶

Returns the p-adic absolute value of self.

This is normalized so that the absolute value of p is 1/p.

INPUT:

- prec -- Integer.  The precision of the real field in
  which the answer is returned.  If None, returns a
  rational for absolutely unramified fields, or a real
  with 53 bits of precision if ramified.

EXAMPLES:

sage: a = Qp(5)(15); a.abs()
1/5
sage: a.abs(53)
0.200000000000000

additive_order(prec)¶

Returns the additive order of self, where self is considered to be zero if it is zero modulo $p^{\mbox{prec}}$ .

INPUT:

self -- a p-adic element
prec -- an integer

OUTPUT:

integer -- the additive order of self

EXAMPLES:

sage: R = Zp(7, 4, 'capped-rel', 'series'); a = R(7^3); a.additive_order(3)
1
sage: a.additive_order(4)
+Infinity
sage: R = Zp(7, 4, 'fixed-mod', 'series'); a = R(7^5); a.additive_order(6)
1

algdep(n)¶

Returns a polynomial of degree at most $n$ which is approximately satisfied by this number. Note that the returned polynomial need not be irreducible, and indeed usually won’t be if this number is a good approximation to an algebraic number of degree less than $n$ .

ALGORITHM:

Uses the PARI C-library algdep command.

INPUT:

- self -- a p-adic element
- n -- an integer

OUTPUT:

polynomial -- degree n polynomial approximately satisfied by self

EXAMPLES:

sage: K = Qp(3,20,'capped-rel','series'); R = Zp(3,20,'capped-rel','series')
sage: a = K(7/19); a
1 + 2*3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + 3^11 + 3^12 + 2*3^15 + 2*3^16 + 3^17 + 2*3^19 + O(3^20)
sage: a.algdep(1)
19*x - 7
sage: K2 = Qp(7,20,'capped-rel')
sage: b = K2.zeta(); b.algdep(2)
x^2 - x + 1
sage: K2 = Qp(11,20,'capped-rel')
sage: b = K2.zeta(); b.algdep(4)
x^4 - x^3 + x^2 - x + 1
sage: a = R(7/19); a
1 + 2*3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + 3^11 + 3^12 + 2*3^15 + 2*3^16 + 3^17 + 2*3^19 + O(3^20)
sage: a.algdep(1)
19*x - 7
sage: R2 = Zp(7,20,'capped-rel')
sage: b = R2.zeta(); b.algdep(2)
x^2 - x + 1
sage: R2 = Zp(11,20,'capped-rel')
sage: b = R2.zeta(); b.algdep(4)
x^4 - x^3 + x^2 - x + 1

algebraic_dependency(n)¶

Returns a polynomial of degree at most $n$ which is approximately satisfied by this number. Note that the returned polynomial need not be irreducible, and indeed usually won’t be if this number is a good approximation to an algebraic number of degree less than $n$ .

ALGORITHM:

Uses the PARI C-library algdep command.

INPUT:

- self -- a p-adic element
- n -- an integer

OUTPUT:

polynomial -- degree n polynomial approximately satisfied by self

EXAMPLES:

sage: K = Qp(3,20,'capped-rel','series'); R = Zp(3,20,'capped-rel','series')
sage: a = K(7/19); a
1 + 2*3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + 3^11 + 3^12 + 2*3^15 + 2*3^16 + 3^17 + 2*3^19 + O(3^20)
sage: a.algebraic_dependency(1)
19*x - 7
sage: K2 = Qp(7,20,'capped-rel')
sage: b = K2.zeta(); b.algebraic_dependency(2)
x^2 - x + 1
sage: K2 = Qp(11,20,'capped-rel')
sage: b = K2.zeta(); b.algebraic_dependency(4)
x^4 - x^3 + x^2 - x + 1
sage: a = R(7/19); a
1 + 2*3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + 3^11 + 3^12 + 2*3^15 + 2*3^16 + 3^17 + 2*3^19 + O(3^20)
sage: a.algebraic_dependency(1)
19*x - 7
sage: R2 = Zp(7,20,'capped-rel')
sage: b = R2.zeta(); b.algebraic_dependency(2)
x^2 - x + 1
sage: R2 = Zp(11,20,'capped-rel')
sage: b = R2.zeta(); b.algebraic_dependency(4)
x^4 - x^3 + x^2 - x + 1

is_square()¶

Returns whether self is a square

INPUT:

self -- a p-adic element

OUTPUT:

boolean -- whether self is a square

EXAMPLES:

sage: R = Zp(3,20,'capped-rel')
sage: R(0).is_square()
True
sage: R(1).is_square()
True
sage: R(2).is_square()
False

TESTS:

sage: R(3).is_square()
False
sage: R(4).is_square()
True
sage: R(6).is_square()
False
sage: R(9).is_square()
True

sage: R2 = Zp(2,20,'capped-rel')
sage: R2(0).is_square()
True
sage: R2(1).is_square()
True
sage: R2(2).is_square()
False
sage: R2(3).is_square()
False
sage: R2(4).is_square()
True
sage: R2(5).is_square()
False
sage: R2(6).is_square()
False
sage: R2(7).is_square()
False
sage: R2(8).is_square()
False
sage: R2(9).is_square()
True

sage: K = Qp(3,20,'capped-rel')
sage: K(0).is_square()
True
sage: K(1).is_square()
True
sage: K(2).is_square()
False
sage: K(3).is_square()
False
sage: K(4).is_square()
True
sage: K(6).is_square()
False
sage: K(9).is_square()
True
sage: K(1/3).is_square()
False
sage: K(1/9).is_square()
True

sage: K2 = Qp(2,20,'capped-rel')
sage: K2(0).is_square()
True
sage: K2(1).is_square()
True
sage: K2(2).is_square()
False
sage: K2(3).is_square()
False
sage: K2(4).is_square()
True
sage: K2(5).is_square()
False
sage: K2(6).is_square()
False
sage: K2(7).is_square()
False
sage: K2(8).is_square()
False
sage: K2(9).is_square()
True
sage: K2(1/2).is_square()
False
sage: K2(1/4).is_square()
True

multiplicative_order(prec=None)¶

Returns the multiplicative order of self, where self is considered to be one if it is one modulo $p^{\mbox{prec}}$ .

INPUT:

self -- a p-adic element
prec -- an integer

OUTPUT:

integer -- the multiplicative order of self

EXAMPLES:

sage: K = Qp(5,20,'capped-rel')
sage: K(-1).multiplicative_order(20)
2
sage: K(1).multiplicative_order(20)
1
sage: K(2).multiplicative_order(20)
+Infinity
sage: K(3).multiplicative_order(20)
+Infinity
sage: K(4).multiplicative_order(20)
+Infinity
sage: K(5).multiplicative_order(20)
+Infinity
sage: K(25).multiplicative_order(20)
+Infinity
sage: K(1/5).multiplicative_order(20)
+Infinity
sage: K(1/25).multiplicative_order(20)
+Infinity
sage: K.zeta().multiplicative_order(20)
4

sage: R = Zp(5,20,'capped-rel')
sage: R(-1).multiplicative_order(20)
2
sage: R(1).multiplicative_order(20)
1
sage: R(2).multiplicative_order(20)
+Infinity
sage: R(3).multiplicative_order(20)
+Infinity
sage: R(4).multiplicative_order(20)
+Infinity
sage: R(5).multiplicative_order(20)
+Infinity
sage: R(25).multiplicative_order(20)
+Infinity
sage: R.zeta().multiplicative_order(20)
4

ordp()¶

Returns the valuation of self, normalized so that the valuation of p is 1

INPUT:

self -- a p-adic element

OUTPUT:

integer -- the valuation of self, normalized so that the valuation of p is 1

EXAMPLES:

sage: R = Zp(5,20,'capped-rel')
sage: R(0).ordp()
+Infinity
sage: R(1).ordp()
0
sage: R(2).ordp()
0
sage: R(5).ordp()
1
sage: R(10).ordp()
1
sage: R(25).ordp()
2
sage: R(50).ordp()
2            
sage: R(1/2).ordp()
0

rational_reconstruction()¶

Returns a rational approximation to this p-adic number

INPUT:

self -- a p-adic element

OUTPUT:

rational -- an approximation to self

EXAMPLES:

sage: R = Zp(5,20,'capped-rel')
sage: for i in range(11):
...       for j in range(1,10):
...           if j == 5:
...               continue
...           assert i/j == R(i/j).rational_reconstruction()

square_root(extend=True, all=False)¶

Returns the square root of this p-adic number

INPUT:

- self -- a p-adic element
- extend -- bool (default: True); if True, return a square root
  in an extension if necessary; if False and no root exists
  in the given ring or field, raise a ValueError
- all -- bool (default: False); if True, return a list of all
  square roots

OUTPUT:

p-adic element -- the square root of this p-adic number

If all = False, the square root chosen is the one whose
reduction mod p is in the range [0, p/2).

EXAMPLES:

sage: R = Zp(3,20,'capped-rel', 'val-unit')
sage: R(0).square_root()
0
sage: R(1).square_root()
1 + O(3^20)
sage: R(2).square_root(extend = False)
...
ValueError: element is not a square
sage: R(4).square_root() == R(-2)
True
sage: R(9).square_root()
3 * 1 + O(3^21)

When p = 2, the precision of the square root is one less
than the input:

sage: R2 = Zp(2,20,'capped-rel')
sage: R2(0).square_root()
0
sage: R2(1).square_root()
1 + O(2^19)
sage: R2(4).square_root()
2 + O(2^20)

sage: R2(9).square_root() == R2(3, 19) or R2(9).square_root() == R2(-3, 19)
True

sage: R2(17).square_root()
1 + 2^3 + 2^5 + 2^6 + 2^7 + 2^9 + 2^10 + 2^13 + 2^16 + 2^17 + O(2^19)

sage: R3 = Zp(5,20,'capped-rel')
sage: R3(0).square_root()
0
sage: R3(1).square_root()
1 + O(5^20)
sage: R3(-1).square_root() == R3.teichmuller(2) or R3(-1).square_root() == R3.teichmuller(3)
True

TESTS:

sage: R = Qp(3,20,'capped-rel')
sage: R(0).square_root()
0
sage: R(1).square_root()
1 + O(3^20)
sage: R(4).square_root() == R(-2)
True
sage: R(9).square_root()
3 + O(3^21)
sage: R(1/9).square_root()
3^-1 + O(3^19)

sage: R2 = Qp(2,20,'capped-rel')
sage: R2(0).square_root()
0
sage: R2(1).square_root()
1 + O(2^19)
sage: R2(4).square_root()
2 + O(2^20)
sage: R2(9).square_root() == R2(3,19) or R2(9).square_root() == R2(-3,19)
True
sage: R2(17).square_root()
1 + 2^3 + 2^5 + 2^6 + 2^7 + 2^9 + 2^10 + 2^13 + 2^16 + 2^17 + O(2^19)

sage: R3 = Qp(5,20,'capped-rel')
sage: R3(0).square_root()
0
sage: R3(1).square_root()
1 + O(5^20)
sage: R3(-1).square_root() == R3.teichmuller(2) or R3(-1).square_root() == R3.teichmuller(3)
True

str(mode=None)¶

Returns a string representation of self.

EXAMPLES:

sage: Zp(5,5,print_mode='bars')(1/3).str()[3:]
'1|3|1|3|2'

val_unit()¶

Returns (self.valuation(), self.unit_part()).

EXAMPLES:

sage: Zp(5,5)(5).val_unit()
(1, 1 + O(5^5))

valuation()¶

Returns the valuation of self.

INPUT:

- self -- a p-adic element

OUTPUT:

- integer -- the valuation of self

EXAMPLES:

sage: R = Zp(17, 4,'capped-rel')
sage: a = R(2*17^2)
sage: a.valuation()
2
sage: R = Zp(5, 4,'capped-rel')
sage: R(0).valuation()
+Infinity

TESTS:

sage: R(1).valuation()
0
sage: R(2).valuation()
0
sage: R(5).valuation()
1
sage: R(10).valuation()
1
sage: R(25).valuation()
2
sage: R(50).valuation()
2            
sage: R = Qp(17, 4)
sage: a = R(2*17^2)
sage: a.valuation()
2
sage: R = Qp(5, 4)
sage: R(0).valuation()
+Infinity
sage: R(1).valuation()
0
sage: R(2).valuation()
0
sage: R(5).valuation()
1
sage: R(10).valuation()
1
sage: R(25).valuation()
2
sage: R(50).valuation()
2            
sage: R(1/2).valuation()
0
sage: R(1/5).valuation()
-1
sage: R(1/10).valuation()
-1
sage: R(1/25).valuation()
-2
sage: R(1/50).valuation()
-2

sage: K.<a> = Qq(25)
sage: K(0).valuation()
+Infinity

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