\(p\)-adic Generic#

A generic superclass for all p-adic parents.

AUTHORS:

David Roe
Genya Zaytman: documentation
David Harvey: doctests
Julian Rueth (2013-03-16): test methods for basic arithmetic

class sage.rings.padics.padic_generic.ResidueLiftingMap#

Bases: Morphism

Lifting map to a p-adic ring or field from its residue field or ring.

These maps must be created using the _create_() method in order to support categories correctly.

EXAMPLES:

sage: from sage.rings.padics.padic_generic import ResidueLiftingMap
sage: R.<a> = Zq(125); k = R.residue_field()
sage: f = ResidueLiftingMap._create_(k, R); f
Lifting morphism:
  From: Finite Field in a0 of size 5^3
  To:   5-adic Unramified Extension Ring in a defined by x^3 + 3*x + 3

class sage.rings.padics.padic_generic.ResidueReductionMap#

Bases: Morphism

Reduction map from a p-adic ring or field to its residue field or ring.

These maps must be created using the _create_() method in order to support categories correctly.

EXAMPLES:

sage: from sage.rings.padics.padic_generic import ResidueReductionMap
sage: R.<a> = Zq(125); k = R.residue_field()
sage: f = ResidueReductionMap._create_(R, k); f
Reduction morphism:
  From: 5-adic Unramified Extension Ring in a defined by x^3 + 3*x + 3
  To:   Finite Field in a0 of size 5^3

is_injective()#

The reduction map is far from injective.

EXAMPLES:

sage: GF(5).convert_map_from(ZpCA(5)).is_injective()
False

is_surjective()#

The reduction map is surjective.

EXAMPLES:

sage: GF(7).convert_map_from(Qp(7)).is_surjective()
True

section()#

Return the section from the residue ring or field back to the p-adic ring or field.

EXAMPLES:

sage: GF(3).convert_map_from(Zp(3)).section()
Lifting morphism:
  From: Finite Field of size 3
  To:   3-adic Ring with capped relative precision 20

sage.rings.padics.padic_generic.local_print_mode(obj, print_options, pos=None, ram_name=None)#

Context manager for safely temporarily changing the print_mode of a p-adic ring/field.

EXAMPLES:

sage: R = Zp(5)
sage: R(45)
4*5 + 5^2 + O(5^21)
sage: with local_print_mode(R, 'val-unit'):
....:     print(R(45))
5 * 9 + O(5^21)

Note

For more documentation see sage.structure.parent_gens.localvars.

class sage.rings.padics.padic_generic.pAdicGeneric(base, p, prec, print_mode, names, element_class, category=None)#

Bases: PrincipalIdealDomain, LocalGeneric

Initialize self.

INPUT:

base – base ring
p – prime
print_mode – dictionary of print options
names – how to print the uniformizer
element_class – the class for elements of this ring

EXAMPLES:

sage: R = Zp(17)  # indirect doctest

characteristic()#

Return the characteristic of self, which is always 0.

EXAMPLES:

sage: R = Zp(3, 10,'fixed-mod'); R.characteristic()
0

extension(modulus, prec=None, names=None, print_mode=None, implementation='FLINT', **kwds)#

Create an extension of this p-adic ring.

EXAMPLES:

sage: k = Qp(5)
sage: R.<x> = k[]
sage: l.<w> = k.extension(x^2-5); l
5-adic Eisenstein Extension Field in w defined by x^2 - 5

sage: F = list(Qp(19)['x'](cyclotomic_polynomial(5)).factor())[0][0]
sage: L = Qp(19).extension(F, names='a')
sage: L
19-adic Unramified Extension Field in a defined by x^2 + 8751674996211859573806383*x + 1

fraction_field(print_mode=None)#

Return the fraction field of this ring or field.

For \(\ZZ_p\), this is the \(p\)-adic field with the same options, and for extensions, it is just the extension of the fraction field of the base determined by the same polynomial.

The fraction field of a capped absolute ring is capped relative, and that of a fixed modulus ring is floating point.

INPUT:

print_mode – (optional) a dictionary containing print options; defaults to the same options as this ring

OUTPUT:

the fraction field of this ring

EXAMPLES:

sage: R = Zp(5, print_mode='digits', show_prec=False)
sage: K = R.fraction_field(); K(1/3)
31313131313131313132
sage: L = R.fraction_field({'max_ram_terms':4}); L(1/3)
doctest:warning
...
DeprecationWarning: Use the change method if you want to change print options in fraction_field()
See https://github.com/sagemath/sage/issues/23227 for details.
3132
sage: U.<a> = Zq(17^4, 6, print_mode='val-unit', print_max_terse_terms=3)
sage: U.fraction_field()
17-adic Unramified Extension Field in a defined by x^4 + 7*x^2 + 10*x + 3
sage: U.fraction_field({"pos":False}) == U.fraction_field()
False

frobenius_endomorphism(n=1)#

Return the \(n\)-th power of the absolute arithmetic Frobeninus endomorphism on this field.

INPUT:

n – an integer (default: 1)

EXAMPLES:

sage: K.<a> = Qq(3^5)
sage: Frob = K.frobenius_endomorphism(); Frob
Frobenius endomorphism on 3-adic Unramified Extension
... lifting a |--> a^3 on the residue field
sage: Frob(a) == a.frobenius()
True

We can specify a power:

sage: K.frobenius_endomorphism(2)
Frobenius endomorphism on 3-adic Unramified Extension
... lifting a |--> a^(3^2) on the residue field

The result is simplified if possible:

sage: K.frobenius_endomorphism(6)
Frobenius endomorphism on 3-adic Unramified Extension
... lifting a |--> a^3 on the residue field
sage: K.frobenius_endomorphism(5)
Identity endomorphism of 3-adic Unramified Extension ...

Comparisons work:

sage: K.frobenius_endomorphism(6) == Frob
True

gens()#

Return a list of generators.

EXAMPLES:

sage: R = Zp(5); R.gens()
[5 + O(5^21)]
sage: Zq(25,names='a').gens()
[a + O(5^20)]
sage: S.<x> = ZZ[]; f = x^5 + 25*x -5; W.<w> = R.ext(f); W.gens()
[w + O(w^101)]

integer_ring(print_mode=None)#

Return the ring of integers of this ring or field.

For \(\QQ_p\), this is the \(p\)-adic ring with the same options, and for extensions, it is just the extension of the ring of integers of the base determined by the same polynomial.

INPUT:

print_mode – (optional) a dictionary containing print options; defaults to the same options as this ring

OUTPUT:

the ring of elements of this field with nonnegative valuation

EXAMPLES:

sage: K = Qp(5, print_mode='digits', show_prec=False)
sage: R = K.integer_ring(); R(1/3)
31313131313131313132
sage: S = K.integer_ring({'max_ram_terms':4}); S(1/3)
doctest:warning
...
DeprecationWarning: Use the change method if you want to change print options in integer_ring()
See https://github.com/sagemath/sage/issues/23227 for details.
3132
sage: U.<a> = Qq(17^4, 6, print_mode='val-unit', print_max_terse_terms=3)
sage: U.integer_ring()
17-adic Unramified Extension Ring in a defined by x^4 + 7*x^2 + 10*x + 3
sage: U.fraction_field({"print_mode":"terse"}) == U.fraction_field()
doctest:warning
...
DeprecationWarning: Use the change method if you want to change print options in fraction_field()
See https://github.com/sagemath/sage/issues/23227 for details.
False

ngens()#

Return the number of generators of self.

We conventionally define this as 1: for base rings, we take a uniformizer as the generator; for extension rings, we take a root of the minimal polynomial defining the extension.

EXAMPLES:

sage: Zp(5).ngens()
1
sage: Zq(25,names='a').ngens()
1

prime()#

Return the prime, ie the characteristic of the residue field.

OUTPUT:

The characteristic of the residue field.

EXAMPLES:

sage: R = Zp(3,5,'fixed-mod')
sage: R.prime()
3

primitive_root_of_unity(n=None, order=False)#

Return a generator of the group of n-th roots of unity in this ring.

INPUT:

n – an integer or None (default: None)
order – a boolean (default: False)

OUTPUT:

A generator of the group of n-th roots of unity. If n is None, a generator of the full group of roots of unity is returned.

If order is True, the order of the above group is returned as well.

EXAMPLES:

sage: R = Zp(5, 10)
sage: zeta = R.primitive_root_of_unity(); zeta
2 + 5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9 + O(5^10)
sage: zeta == R.teichmuller(2)
True

Now we consider an example with non trivial p-th roots of unity:

sage: W = Zp(3, 2)
sage: S.<x> = W[]
sage: R.<pi> = W.extension((x+1)^6 + (x+1)^3 + 1)

sage: zeta, order = R.primitive_root_of_unity(order=True)
sage: zeta
2 + 2*pi + 2*pi^3 + 2*pi^7 + 2*pi^8 + 2*pi^9 + pi^11 + O(pi^12)
sage: order
18
sage: zeta.multiplicative_order()
18

sage: zeta, order = R.primitive_root_of_unity(24, order=True)
sage: zeta
2 + pi^3 + 2*pi^7 + 2*pi^8 + 2*pi^10 + 2*pi^11 + O(pi^12)
sage: order   # equal to gcd(18,24)
6
sage: zeta.multiplicative_order()
6

print_mode()#

Return the current print mode as a string.

EXAMPLES:

sage: R = Qp(7,5, 'capped-rel')
sage: R.print_mode()
'series'

residue_characteristic()#

Return the prime, i.e., the characteristic of the residue field.

OUTPUT:

The characteristic of the residue field.

EXAMPLES:

sage: R = Zp(3,5,'fixed-mod')
sage: R.residue_characteristic()
3

residue_class_field()#

Return the residue class field.

EXAMPLES:

sage: R = Zp(3,5,'fixed-mod')
sage: k = R.residue_class_field()
sage: k
Finite Field of size 3

residue_field()#

Return the residue class field.

EXAMPLES:

sage: R = Zp(3,5,'fixed-mod')
sage: k = R.residue_field()
sage: k
Finite Field of size 3

residue_ring(n)#

Return the quotient of the ring of integers by the n-th power of the maximal ideal.

EXAMPLES:

sage: R = Zp(11)
sage: R.residue_ring(3)
Ring of integers modulo 1331

residue_system()#

Return a list of elements representing all the residue classes.

EXAMPLES:

sage: R = Zp(3, 5,'fixed-mod')
sage: R.residue_system()
[0, 1, 2]

roots_of_unity(n=None)#

Return all the n-th roots of unity in this ring.

INPUT:

n – an integer or None (default: None); if None, the full group of roots of unity is returned

EXAMPLES:

sage: R = Zp(5, 10)
sage: roots = R.roots_of_unity(); roots
[1 + O(5^10),
 2 + 5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9 + O(5^10),
 4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + O(5^10),
 3 + 3*5 + 2*5^2 + 3*5^3 + 5^4 + 2*5^6 + 5^7 + 4*5^8 + 5^9 + O(5^10)]

sage: R.roots_of_unity(10)
[1 + O(5^10),
 4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + O(5^10)]

In this case, the roots of unity are the Teichmüller representatives:

sage: R.teichmuller_system()
[1 + O(5^10),
 2 + 5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9 + O(5^10),
 3 + 3*5 + 2*5^2 + 3*5^3 + 5^4 + 2*5^6 + 5^7 + 4*5^8 + 5^9 + O(5^10),
 4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + O(5^10)]

In general, there might be more roots of unity (it happens when the ring has non trivial p-th roots of unity):

sage: W.<a> = Zq(3^2, 2)
sage: S.<x> = W[]
sage: R.<pi> = W.extension((x+1)^2 + (x+1) + 1)

sage: roots = R.roots_of_unity(); roots
[1 + O(pi^4),
 a + 2*a*pi + 2*a*pi^2 + a*pi^3 + O(pi^4),
 ...
 1 + pi + O(pi^4),
 a + a*pi^2 + 2*a*pi^3 + O(pi^4),
 ...
 1 + 2*pi + pi^2 + O(pi^4),
 a + a*pi + a*pi^2 + O(pi^4),
 ...]
sage: len(roots)
24

We check that the logarithm of each root of unity vanishes:

sage: for root in roots:
....:     if root.log() != 0:
....:         raise ValueError

some_elements()#

Return a list of elements in this ring.

This is typically used for running generic tests (see TestSuite).

EXAMPLES:

sage: Zp(2,4).some_elements()
[0, 1 + O(2^4), 2 + O(2^5), 1 + 2^2 + 2^3 + O(2^4), 2 + 2^2 + 2^3 + 2^4 + O(2^5)]

teichmuller(x, prec=None)#

Return the Teichmüller representative of x.

x – something that can be cast into self

OUTPUT:

the Teichmüller lift of x

EXAMPLES:

sage: R = Zp(5, 10, 'capped-rel', 'series')
sage: R.teichmuller(2)
2 + 5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9 + O(5^10)
sage: R = Qp(5, 10,'capped-rel','series')
sage: R.teichmuller(2)
2 + 5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9 + O(5^10)
sage: R = Zp(5, 10, 'capped-abs', 'series')
sage: R.teichmuller(2)
2 + 5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9 + O(5^10)
sage: R = Zp(5, 10, 'fixed-mod', 'series')
sage: R.teichmuller(2)
2 + 5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9
sage: R = Zp(5,5)
sage: S.<x> = R[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f)
sage: y = W.teichmuller(3); y
3 + 3*w^5 + w^7 + 2*w^9 + 2*w^10 + 4*w^11 + w^12 + 2*w^13 + 3*w^15
 + 2*w^16 + 3*w^17 + w^18 + 3*w^19 + 3*w^20 + 2*w^21 + 2*w^22
 + 3*w^23 + 4*w^24 + O(w^25)
sage: y^5 == y
True
sage: g = x^3 + 3*x + 3
sage: A.<a> = R.ext(g)
sage: b = A.teichmuller(1 + 2*a - a^2); b
(4*a^2 + 2*a + 1) + 2*a*5 + (3*a^2 + 1)*5^2 + (a + 4)*5^3
 + (a^2 + a + 1)*5^4 + O(5^5)
sage: b^125 == b
True

We check that github issue #23736 is resolved:

sage: R.teichmuller(GF(5)(2))
2 + 5 + 2*5^2 + 5^3 + 3*5^4 + O(5^5)

AUTHORS:

Initial version: David Roe
Quadratic time version: Kiran Kedlaya <kedlaya@math.mit.edu> (2007-03-27)

teichmuller_system()#

Return a set of Teichmüller representatives for the invertible elements of \(\ZZ / p\ZZ\).

OUTPUT:

A list of Teichmüller representatives for the invertible elements of \(\ZZ / p\ZZ\).

EXAMPLES:

sage: R = Zp(3, 5,'fixed-mod', 'terse')
sage: R.teichmuller_system()
[1, 242]

Check that github issue #20457 is fixed:

sage: F.<a> = Qq(5^2,6)
sage: F.teichmuller_system()[3]
(2*a + 2) + (4*a + 1)*5 + 4*5^2 + (2*a + 1)*5^3 + (4*a + 1)*5^4 + (2*a + 3)*5^5 + O(5^6)

Note

Should this return 0 as well?

uniformizer_pow(n)#

Return \(p^n\), as an element of self.

If n is infinity, returns 0.

EXAMPLES:

sage: R = Zp(3, 5, 'fixed-mod')
sage: R.uniformizer_pow(3)
3^3
sage: R.uniformizer_pow(infinity)
0

valuation()#

Return the \(p\)-adic valuation on this ring.

OUTPUT:

A valuation that is normalized such that the rational prime \(p\) has valuation 1.

EXAMPLES:

sage: K = Qp(3)
sage: R.<a> = K[]
sage: L.<a> = K.extension(a^3 - 3)
sage: v = L.valuation(); v
3-adic valuation
sage: v(3)
1
sage: L(3).valuation()
3

The normalization is chosen such that the valuation restricts to the valuation on the base ring:

sage: v(3) == K.valuation()(3)
True
sage: v.restriction(K) == K.valuation()
True