\(p\)-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[source]

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()                                    # needs sage.libs.ntl
sage: f = ResidueLiftingMap._create_(k, R); f                                   # needs sage.libs.ntl
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

>>> from sage.all import *
>>> from sage.rings.padics.padic_generic import ResidueLiftingMap
>>> R = Zq(Integer(125), names=('a',)); (a,) = R._first_ngens(1); k = R.residue_field()                                    # needs sage.libs.ntl
>>> f = ResidueLiftingMap._create_(k, R); f                                   # needs sage.libs.ntl
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[source]

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()                                    # needs sage.libs.ntl
sage: f = ResidueReductionMap._create_(R, k); f                                 # needs sage.libs.ntl
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

>>> from sage.all import *
>>> from sage.rings.padics.padic_generic import ResidueReductionMap
>>> R = Zq(Integer(125), names=('a',)); (a,) = R._first_ngens(1); k = R.residue_field()                                    # needs sage.libs.ntl
>>> f = ResidueReductionMap._create_(R, k); f                                 # needs sage.libs.ntl
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()[source]

The reduction map is far from injective.

EXAMPLES:

sage: GF(5).convert_map_from(ZpCA(5)).is_injective()                        # needs sage.rings.finite_rings
False

>>> from sage.all import *
>>> GF(Integer(5)).convert_map_from(ZpCA(Integer(5))).is_injective()                        # needs sage.rings.finite_rings
False
is_surjective()[source]

The reduction map is surjective.

EXAMPLES:

sage: GF(7).convert_map_from(Qp(7)).is_surjective()                         # needs sage.rings.finite_rings
True

>>> from sage.all import *
>>> GF(Integer(7)).convert_map_from(Qp(Integer(7))).is_surjective()                         # needs sage.rings.finite_rings
True
section()[source]

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()                               # needs sage.rings.finite_rings
Lifting morphism:
  From: Finite Field of size 3
  To:   3-adic Ring with capped relative precision 20

>>> from sage.all import *
>>> GF(Integer(3)).convert_map_from(Zp(Integer(3))).section()                               # needs sage.rings.finite_rings
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)[source]

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)

>>> from sage.all import *
>>> R = Zp(Integer(5))
>>> R(Integer(45))
4*5 + 5^2 + O(5^21)
>>> with local_print_mode(R, 'val-unit'):
...     print(R(Integer(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)[source]

Bases: 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

>>> from sage.all import *
>>> R = Zp(Integer(17))  # indirect doctest
characteristic()[source]

Return the characteristic of self, which is always 0.

EXAMPLES:

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

>>> from sage.all import *
>>> R = Zp(Integer(3), Integer(10),'fixed-mod'); R.characteristic()
0
extension(modulus, prec=None, names=None, print_mode=None, implementation='FLINT', **kwds)[source]

Create an extension of this \(p\)-adic ring.

EXAMPLES:

sage: # needs sage.libs.ntl
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'); L
19-adic Unramified Extension Field in a defined by x^2 + 8751674996211859573806383*x + 1

>>> from sage.all import *
>>> # needs sage.libs.ntl
>>> k = Qp(Integer(5))
>>> R = k['x']; (x,) = R._first_ngens(1)
>>> l = k.extension(x**Integer(2) - Integer(5), names=('w',)); (w,) = l._first_ngens(1); l
5-adic Eisenstein Extension Field in w defined by x^2 - 5
>>> F = list(Qp(Integer(19))['x'](cyclotomic_polynomial(Integer(5))).factor())[Integer(0)][Integer(0)]
>>> L = Qp(Integer(19)).extension(F, names='a'); L
19-adic Unramified Extension Field in a defined by x^2 + 8751674996211859573806383*x + 1
fraction_field(print_mode=None)[source]

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)   # needs sage.libs.ntl
sage: U.fraction_field()                                                    # needs sage.libs.ntl
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()                 # needs sage.libs.ntl
False

>>> from sage.all import *
>>> R = Zp(Integer(5), print_mode='digits', show_prec=False)
>>> K = R.fraction_field(); K(Integer(1)/Integer(3))
31313131313131313132
>>> L = R.fraction_field({'max_ram_terms':Integer(4)}); L(Integer(1)/Integer(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
>>> U = Zq(Integer(17)**Integer(4), Integer(6), print_mode='val-unit', print_max_terse_terms=Integer(3), names=('a',)); (a,) = U._first_ngens(1)# needs sage.libs.ntl
>>> U.fraction_field()                                                    # needs sage.libs.ntl
17-adic Unramified Extension Field in a defined by x^4 + 7*x^2 + 10*x + 3
>>> U.fraction_field({"pos":False}) == U.fraction_field()                 # needs sage.libs.ntl
False
frobenius_endomorphism(n=1)[source]

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

INPUT:

  • n – integer (default: 1)

EXAMPLES:

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

>>> from sage.all import *
>>> K = Qq(Integer(3)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.libs.ntl
>>> Frob = K.frobenius_endomorphism(); Frob                               # needs sage.libs.ntl
Frobenius endomorphism on 3-adic Unramified Extension
... lifting a |--> a^3 on the residue field
>>> Frob(a) == a.frobenius()                                              # needs sage.libs.ntl
True

We can specify a power:

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

>>> from sage.all import *
>>> K.frobenius_endomorphism(Integer(2))                                           # needs sage.libs.ntl
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)                                           # needs sage.libs.ntl
Frobenius endomorphism on 3-adic Unramified Extension
... lifting a |--> a^3 on the residue field
sage: K.frobenius_endomorphism(5)                                           # needs sage.libs.ntl
Identity endomorphism of 3-adic Unramified Extension ...

>>> from sage.all import *
>>> K.frobenius_endomorphism(Integer(6))                                           # needs sage.libs.ntl
Frobenius endomorphism on 3-adic Unramified Extension
... lifting a |--> a^3 on the residue field
>>> K.frobenius_endomorphism(Integer(5))                                           # needs sage.libs.ntl
Identity endomorphism of 3-adic Unramified Extension ...

Comparisons work:

sage: K.frobenius_endomorphism(6) == Frob                                   # needs sage.libs.ntl
True

>>> from sage.all import *
>>> K.frobenius_endomorphism(Integer(6)) == Frob                                   # needs sage.libs.ntl
True
gens()[source]

Return a tuple of generators.

EXAMPLES:

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

>>> from sage.all import *
>>> R = Zp(Integer(5)); R.gens()
(5 + O(5^21),)
>>> Zq(Integer(25),names='a').gens()                                               # needs sage.libs.ntl
(a + O(5^20),)
>>> S = ZZ['x']; (x,) = S._first_ngens(1); f = x**Integer(5) + Integer(25)*x -Integer(5); W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1); W.gens()           # needs sage.libs.ntl
(w + O(w^101),)
integer_ring(print_mode=None)[source]

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)   # needs sage.libs.ntl
sage: U.integer_ring()                                                      # needs sage.libs.ntl
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()        # needs sage.libs.ntl
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

>>> from sage.all import *
>>> K = Qp(Integer(5), print_mode='digits', show_prec=False)
>>> R = K.integer_ring(); R(Integer(1)/Integer(3))
31313131313131313132
>>> S = K.integer_ring({'max_ram_terms':Integer(4)}); S(Integer(1)/Integer(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
>>> U = Qq(Integer(17)**Integer(4), Integer(6), print_mode='val-unit', print_max_terse_terms=Integer(3), names=('a',)); (a,) = U._first_ngens(1)# needs sage.libs.ntl
>>> U.integer_ring()                                                      # needs sage.libs.ntl
17-adic Unramified Extension Ring in a defined by x^4 + 7*x^2 + 10*x + 3
>>> U.fraction_field({"print_mode":"terse"}) == U.fraction_field()        # needs sage.libs.ntl
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()[source]

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()                                              # needs sage.libs.ntl
1

>>> from sage.all import *
>>> Zp(Integer(5)).ngens()
1
>>> Zq(Integer(25),names='a').ngens()                                              # needs sage.libs.ntl
1
prime()[source]

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

>>> from sage.all import *
>>> R = Zp(Integer(3),Integer(5),'fixed-mod')
>>> R.prime()
3
primitive_root_of_unity(n=None, order=False)[source]

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

INPUT:

  • n – integer or None (default: None)

  • order – 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

>>> from sage.all import *
>>> R = Zp(Integer(5), Integer(10))
>>> 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)
>>> zeta == R.teichmuller(Integer(2))
True

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

sage: # needs sage.libs.ntl
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

>>> from sage.all import *
>>> # needs sage.libs.ntl
>>> W = Zp(Integer(3), Integer(2))
>>> S = W['x']; (x,) = S._first_ngens(1)
>>> R = W.extension((x+Integer(1))**Integer(6) + (x+Integer(1))**Integer(3) + Integer(1), names=('pi',)); (pi,) = R._first_ngens(1)
>>> zeta, order = R.primitive_root_of_unity(order=True)
>>> zeta
2 + 2*pi + 2*pi^3 + 2*pi^7 + 2*pi^8 + 2*pi^9 + pi^11 + O(pi^12)
>>> order
18
>>> zeta.multiplicative_order()
18
>>> zeta, order = R.primitive_root_of_unity(Integer(24), order=True)
>>> zeta
2 + pi^3 + 2*pi^7 + 2*pi^8 + 2*pi^10 + 2*pi^11 + O(pi^12)
>>> order   # equal to gcd(18,24)
6
>>> zeta.multiplicative_order()
6
print_mode()[source]

Return the current print mode as a string.

EXAMPLES:

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

>>> from sage.all import *
>>> R = Qp(Integer(7),Integer(5), 'capped-rel')
>>> R.print_mode()
'series'
residue_characteristic()[source]

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

>>> from sage.all import *
>>> R = Zp(Integer(3),Integer(5),'fixed-mod')
>>> R.residue_characteristic()
3
residue_class_field()[source]

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

>>> from sage.all import *
>>> R = Zp(Integer(3),Integer(5),'fixed-mod')
>>> k = R.residue_class_field()
>>> k
Finite Field of size 3
residue_field()[source]

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

>>> from sage.all import *
>>> R = Zp(Integer(3),Integer(5),'fixed-mod')
>>> k = R.residue_field()
>>> k
Finite Field of size 3
residue_ring(n)[source]

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

>>> from sage.all import *
>>> R = Zp(Integer(11))
>>> R.residue_ring(Integer(3))
Ring of integers modulo 1331
residue_system()[source]

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]

>>> from sage.all import *
>>> R = Zp(Integer(3), Integer(5),'fixed-mod')
>>> R.residue_system()
[0, 1, 2]
roots_of_unity(n=None)[source]

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

INPUT:

  • n – 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)]

>>> from sage.all import *
>>> R = Zp(Integer(5), Integer(10))
>>> 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)]

>>> R.roots_of_unity(Integer(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)]

>>> from sage.all import *
>>> 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: # needs sage.libs.ntl
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

>>> from sage.all import *
>>> # needs sage.libs.ntl
>>> W = Zq(Integer(3)**Integer(2), Integer(2), names=('a',)); (a,) = W._first_ngens(1)
>>> S = W['x']; (x,) = S._first_ngens(1)
>>> R = W.extension((x+Integer(1))**Integer(2) + (x+Integer(1)) + Integer(1), names=('pi',)); (pi,) = R._first_ngens(1)
>>> 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),
 ...]
>>> len(roots)
24

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

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

>>> from sage.all import *
>>> # needs sage.libs.ntl
>>> for root in roots:
...     if root.log() != Integer(0):
...         raise ValueError
some_elements()[source]

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)]

>>> from sage.all import *
>>> Zp(Integer(2),Integer(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)[source]

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: # needs sage.libs.ntl
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

>>> from sage.all import *
>>> R = Zp(Integer(5), Integer(10), 'capped-rel', 'series')
>>> R.teichmuller(Integer(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)
>>> R = Qp(Integer(5), Integer(10),'capped-rel','series')
>>> R.teichmuller(Integer(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)
>>> R = Zp(Integer(5), Integer(10), 'capped-abs', 'series')
>>> R.teichmuller(Integer(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)
>>> R = Zp(Integer(5), Integer(10), 'fixed-mod', 'series')
>>> R.teichmuller(Integer(2))
2 + 5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9

>>> # needs sage.libs.ntl
>>> R = Zp(Integer(5),Integer(5))
>>> S = R['x']; (x,) = S._first_ngens(1)
>>> f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) +Integer(125)*x - Integer(5)
>>> W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1)
>>> y = W.teichmuller(Integer(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)
>>> y**Integer(5) == y
True
>>> g = x**Integer(3) + Integer(3)*x + Integer(3)
>>> A = R.ext(g, names=('a',)); (a,) = A._first_ngens(1)
>>> b = A.teichmuller(Integer(1) + Integer(2)*a - a**Integer(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)
>>> b**Integer(125) == b
True

We check that Issue #23736 is resolved:

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

>>> from sage.all import *
>>> # needs sage.libs.ntl
>>> R.teichmuller(GF(Integer(5))(Integer(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()[source]

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]

>>> from sage.all import *
>>> R = Zp(Integer(3), Integer(5),'fixed-mod', 'terse')
>>> R.teichmuller_system()
[1, 242]

Check that Issue #20457 is fixed:

sage: F.<a> = Qq(5^2,6)                                                     # needs sage.libs.ntl
sage: F.teichmuller_system()[3]                                             # needs sage.libs.ntl
(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)

>>> from sage.all import *
>>> F = Qq(Integer(5)**Integer(2),Integer(6), names=('a',)); (a,) = F._first_ngens(1)# needs sage.libs.ntl
>>> F.teichmuller_system()[Integer(3)]                                             # needs sage.libs.ntl
(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)[source]

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

>>> from sage.all import *
>>> R = Zp(Integer(3), Integer(5), 'fixed-mod')
>>> R.uniformizer_pow(Integer(3))
3^3
>>> R.uniformizer_pow(infinity)
0
valuation()[source]

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: # needs sage.libs.ntl
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

>>> from sage.all import *
>>> # needs sage.libs.ntl
>>> K = Qp(Integer(3))
>>> R = K['a']; (a,) = R._first_ngens(1)
>>> L = K.extension(a**Integer(3) - Integer(3), names=('a',)); (a,) = L._first_ngens(1)
>>> v = L.valuation(); v
3-adic valuation
>>> v(Integer(3))
1
>>> L(Integer(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)                                              # needs sage.libs.ntl
True
sage: v.restriction(K) == K.valuation()                                     # needs sage.libs.ntl
True

>>> from sage.all import *
>>> v(Integer(3)) == K.valuation()(Integer(3))                                              # needs sage.libs.ntl
True
>>> v.restriction(K) == K.valuation()                                     # needs sage.libs.ntl
True

See also

NumberField_generic.valuation(), Order.valuation()