# $$p$$-adic Extension Generic#

A common superclass for all extensions of Qp and Zp.

AUTHORS:

• David Roe

Bases: Morphism

Conversion map between p-adic rings/fields with the same defining polynomial.

INPUT:

• R – a p-adic extension ring or field.

• S – a p-adic extension ring or field with the same defining polynomial.

EXAMPLES:

sage: R.<a> = Zq(125, print_mode='terse')
sage: S = R.change(prec = 15, type='floating-point')
sage: a - 1
95367431640624 + a + O(5^20)
sage: S(a - 1)
30517578124 + a + O(5^15)

>>> from sage.all import *
>>> R = Zq(Integer(125), print_mode='terse', names=('a',)); (a,) = R._first_ngens(1)
>>> S = R.change(prec = Integer(15), type='floating-point')
>>> a - Integer(1)
95367431640624 + a + O(5^20)
>>> S(a - Integer(1))
30517578124 + a + O(5^15)

sage: R.<a> = Zq(125, print_mode='terse')
sage: S = R.change(prec = 15, type='floating-point')
sage: f = S.convert_map_from(R)
sage: TestSuite(f).run()

>>> from sage.all import *
>>> R = Zq(Integer(125), print_mode='terse', names=('a',)); (a,) = R._first_ngens(1)
>>> S = R.change(prec = Integer(15), type='floating-point')
>>> f = S.convert_map_from(R)
>>> TestSuite(f).run()


The isomorphism from the underlying module of a one-step p-adic extension to the extension.

EXAMPLES:

sage: K.<a> = Qq(125)
sage: V, fr, to = K.free_module()
sage: TestSuite(fr).run(skip=['_test_nonzero_equal'])  # skipped since Qq(125) doesn't have dimension()

>>> from sage.all import *
>>> K = Qq(Integer(125), names=('a',)); (a,) = K._first_ngens(1)
>>> V, fr, to = K.free_module()
>>> TestSuite(fr).run(skip=['_test_nonzero_equal'])  # skipped since Qq(125) doesn't have dimension()


The isomorphism from the underlying module of a two-step p-adic extension to the extension.

EXAMPLES:

sage: K.<a> = Qq(125)
sage: R.<x> = ZZ[]
sage: L.<b> = K.extension(x^2 - 5*x + 5)
sage: V, fr, to = L.free_module(base=Qp(5))
sage: TestSuite(fr).run(skip=['_test_nonzero_equal'])  # skipped since L doesn't have dimension()

>>> from sage.all import *
>>> K = Qq(Integer(125), names=('a',)); (a,) = K._first_ngens(1)
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> L = K.extension(x**Integer(2) - Integer(5)*x + Integer(5), names=('b',)); (b,) = L._first_ngens(1)
>>> V, fr, to = L.free_module(base=Qp(Integer(5)))
>>> TestSuite(fr).run(skip=['_test_nonzero_equal'])  # skipped since L doesn't have dimension()


The isomorphism from a one-step p-adic extension to its underlying free module

EXAMPLES:

sage: K.<a> = Qq(125)
sage: V, fr, to = K.free_module()
sage: TestSuite(to).run()

>>> from sage.all import *
>>> K = Qq(Integer(125), names=('a',)); (a,) = K._first_ngens(1)
>>> V, fr, to = K.free_module()
>>> TestSuite(to).run()


The isomorphism from a two-step p-adic extension to its underlying free module

EXAMPLES:

sage: K.<a> = Qq(125)
sage: R.<x> = ZZ[]
sage: L.<b> = K.extension(x^2 - 5*x + 5)
sage: V, fr, to = L.free_module(base=Qp(5))
sage: TestSuite(to).run()

>>> from sage.all import *
>>> K = Qq(Integer(125), names=('a',)); (a,) = K._first_ngens(1)
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> L = K.extension(x**Integer(2) - Integer(5)*x + Integer(5), names=('b',)); (b,) = L._first_ngens(1)
>>> V, fr, to = L.free_module(base=Qp(Integer(5)))
>>> TestSuite(to).run()


Bases: pAdicGeneric

Initialization

EXAMPLES:

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)  # indirect doctest

>>> from sage.all import *
>>> 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)# indirect doctest

construction(forbid_frac_field=False)[source]#

Returns the functorial construction of this ring, namely, the algebraic extension of the base ring defined by the given polynomial.

Also preserves other information that makes this ring unique (e.g. precision, rounding, print mode).

INPUT:

EXAMPLES:

sage: R.<a> = Zq(25, 8, print_mode='val-unit')
sage: c, R0 = R.construction(); R0
5-adic Ring with capped relative precision 8
sage: c(R0)
5-adic Unramified Extension Ring in a defined by x^2 + 4*x + 2
sage: c(R0) == R
True

>>> from sage.all import *
>>> R = Zq(Integer(25), Integer(8), print_mode='val-unit', names=('a',)); (a,) = R._first_ngens(1)
>>> c, R0 = R.construction(); R0
5-adic Ring with capped relative precision 8
>>> c(R0)
5-adic Unramified Extension Ring in a defined by x^2 + 4*x + 2
>>> c(R0) == R
True


For a field, by default we return a fraction field functor.

sage: K.<a> = Qq(25, 8)
sage: c, R = K.construction(); R
5-adic Unramified Extension Ring in a defined by x^2 + 4*x + 2
sage: c
FractionField

>>> from sage.all import *
>>> K = Qq(Integer(25), Integer(8), names=('a',)); (a,) = K._first_ngens(1)
>>> c, R = K.construction(); R
5-adic Unramified Extension Ring in a defined by x^2 + 4*x + 2
>>> c
FractionField


If you prefer an extension functor, you can use the forbit_frac_field keyword:

sage: c, R = K.construction(forbid_frac_field=True); R
5-adic Field with capped relative precision 8
sage: c
AlgebraicExtensionFunctor
sage: c(R) is K
True

>>> from sage.all import *
>>> c, R = K.construction(forbid_frac_field=True); R
5-adic Field with capped relative precision 8
>>> c
AlgebraicExtensionFunctor
>>> c(R) is K
True

defining_polynomial(var=None, exact=False)[source]#

Returns the polynomial defining this extension.

INPUT:

• var – string (default: 'x'), the name of the variable

• exact – boolean (default False), whether to return the underlying exact

defining polynomial rather than the one with coefficients in the base ring.

EXAMPLES:

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: W.defining_polynomial()
(1 + O(5^5))*x^5 + O(5^6)*x^4 + (3*5^2 + O(5^6))*x^3 + (2*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + O(5^6))*x^2 + (5^3 + O(5^6))*x + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + O(5^6)
sage: W.defining_polynomial(exact=True)
x^5 + 75*x^3 - 15*x^2 + 125*x - 5

sage: W.defining_polynomial(var='y', exact=True)
y^5 + 75*y^3 - 15*y^2 + 125*y - 5

>>> from sage.all import *
>>> 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)
>>> W.defining_polynomial()
(1 + O(5^5))*x^5 + O(5^6)*x^4 + (3*5^2 + O(5^6))*x^3 + (2*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + O(5^6))*x^2 + (5^3 + O(5^6))*x + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + O(5^6)
>>> W.defining_polynomial(exact=True)
x^5 + 75*x^3 - 15*x^2 + 125*x - 5

>>> W.defining_polynomial(var='y', exact=True)
y^5 + 75*y^3 - 15*y^2 + 125*y - 5

exact_field()[source]#

Return a number field with the same defining polynomial.

Note that this method always returns a field, even for a $$p$$-adic ring.

EXAMPLES:

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: W.exact_field()
Number Field in w with defining polynomial x^5 + 75*x^3 - 15*x^2 + 125*x - 5

>>> from sage.all import *
>>> 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)
>>> W.exact_field()
Number Field in w with defining polynomial x^5 + 75*x^3 - 15*x^2 + 125*x - 5

exact_ring()[source]#

Return the order with the same defining polynomial.

Will raise a ValueError if the coefficients of the defining polynomial are not integral.

EXAMPLES:

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: W.exact_ring()
Order generated by w in Number Field in w with defining polynomial x^5 + 75*x^3 - 15*x^2 + 125*x - 5

sage: T = Zp(5,5)
sage: U.<z> = T[]
sage: g = 2*z^4 + 1
sage: V.<v> = T.ext(g)
sage: V.exact_ring()
Traceback (most recent call last):
...
ValueError: each generator must be integral

>>> from sage.all import *
>>> 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)
>>> W.exact_ring()
Order generated by w in Number Field in w with defining polynomial x^5 + 75*x^3 - 15*x^2 + 125*x - 5

>>> T = Zp(Integer(5),Integer(5))
>>> U = T['z']; (z,) = U._first_ngens(1)
>>> g = Integer(2)*z**Integer(4) + Integer(1)
>>> V = T.ext(g, names=('v',)); (v,) = V._first_ngens(1)
>>> V.exact_ring()
Traceback (most recent call last):
...
ValueError: each generator must be integral

free_module(base=None, basis=None, map=True)[source]#

Return a free module $$V$$ over a specified base ring together with maps to and from $$V$$.

INPUT:

• base – a subring $$R$$ so that this ring/field is isomorphic to a finite-rank free $$R$$-module $$V$$

• basis – a basis for this ring/field over the base

• map – boolean (default True), whether to return $$R$$-linear maps to and from $$V$$

OUTPUT:

• A finite-rank free $$R$$-module $$V$$

• An $$R$$-module isomorphism from $$V$$ to this ring/field (only included if map is True)

• An $$R$$-module isomorphism from this ring/field to $$V$$ (only included if map is True)

EXAMPLES:

sage: R.<x> = ZZ[]
sage: K.<a> = Qq(125)
sage: L.<pi> = K.extension(x^2-5)
sage: V, from_V, to_V = K.free_module()
sage: W, from_W, to_W = L.free_module()
sage: W0, from_W0, to_W0 = L.free_module(base=Qp(5))
sage: to_V(a + O(5^7))
(O(5^7), 1 + O(5^7), O(5^7))
sage: to_W(a)
(a + O(5^20), O(5^20))
sage: to_W0(a + O(5^7))
(O(5^7), 1 + O(5^7), O(5^7), O(5^7), O(5^7), O(5^7))
sage: to_W(pi)
(O(5^21), 1 + O(5^20))
sage: to_W0(pi + O(pi^11))
(O(5^6), O(5^6), O(5^6), 1 + O(5^5), O(5^5), O(5^5))

sage: X, from_X, to_X = K.free_module(K)
sage: to_X(a)
(a + O(5^20))

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> K = Qq(Integer(125), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.extension(x**Integer(2)-Integer(5), names=('pi',)); (pi,) = L._first_ngens(1)
>>> V, from_V, to_V = K.free_module()
>>> W, from_W, to_W = L.free_module()
>>> W0, from_W0, to_W0 = L.free_module(base=Qp(Integer(5)))
>>> to_V(a + O(Integer(5)**Integer(7)))
(O(5^7), 1 + O(5^7), O(5^7))
>>> to_W(a)
(a + O(5^20), O(5^20))
>>> to_W0(a + O(Integer(5)**Integer(7)))
(O(5^7), 1 + O(5^7), O(5^7), O(5^7), O(5^7), O(5^7))
>>> to_W(pi)
(O(5^21), 1 + O(5^20))
>>> to_W0(pi + O(pi**Integer(11)))
(O(5^6), O(5^6), O(5^6), 1 + O(5^5), O(5^5), O(5^5))

>>> X, from_X, to_X = K.free_module(K)
>>> to_X(a)
(a + O(5^20))

ground_ring()[source]#

Returns the ring of which this ring is an extension.

EXAMPLES:

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: W.ground_ring()
5-adic Ring with capped relative precision 5

>>> from sage.all import *
>>> 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)
>>> W.ground_ring()
5-adic Ring with capped relative precision 5

ground_ring_of_tower()[source]#

Returns the p-adic base ring of which this is ultimately an extension.

Currently this function is identical to ground_ring(), since relative extensions have not yet been implemented.

EXAMPLES:

sage: Qq(27,30,names='a').ground_ring_of_tower()
3-adic Field with capped relative precision 30

>>> from sage.all import *
>>> Qq(Integer(27),Integer(30),names='a').ground_ring_of_tower()
3-adic Field with capped relative precision 30

modulus(exact=False)[source]#

Returns the polynomial defining this extension.

INPUT:

• exact – boolean (default False), whether to return the underlying exact

defining polynomial rather than the one with coefficients in the base ring.

EXAMPLES:

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: W.modulus()
(1 + O(5^5))*x^5 + O(5^6)*x^4 + (3*5^2 + O(5^6))*x^3 + (2*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + O(5^6))*x^2 + (5^3 + O(5^6))*x + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + O(5^6)
sage: W.modulus(exact=True)
x^5 + 75*x^3 - 15*x^2 + 125*x - 5

>>> from sage.all import *
>>> 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)
>>> W.modulus()
(1 + O(5^5))*x^5 + O(5^6)*x^4 + (3*5^2 + O(5^6))*x^3 + (2*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + O(5^6))*x^2 + (5^3 + O(5^6))*x + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + O(5^6)
>>> W.modulus(exact=True)
x^5 + 75*x^3 - 15*x^2 + 125*x - 5

polynomial_ring()[source]#

Returns the polynomial ring of which this is a quotient.

EXAMPLES:

sage: Qq(27,30,names='a').polynomial_ring()
Univariate Polynomial Ring in x over 3-adic Field with capped relative precision 30

>>> from sage.all import *
>>> Qq(Integer(27),Integer(30),names='a').polynomial_ring()
Univariate Polynomial Ring in x over 3-adic Field with capped relative precision 30

random_element()[source]#

Return a random element of self.

This is done by picking a random element of the ground ring self.degree() times, then treating those elements as coefficients of a polynomial in self.gen().

EXAMPLES:

sage: R.<a> = Zq(125, 5)
sage: R.random_element().parent() is R
True
sage: R = Zp(5,3); S.<x> = ZZ[]; f = x^5 + 25*x^2 - 5; W.<w> = R.ext(f)
sage: W.random_element().parent() is W
True

>>> from sage.all import *
>>> R = Zq(Integer(125), Integer(5), names=('a',)); (a,) = R._first_ngens(1)
>>> R.random_element().parent() is R
True
>>> R = Zp(Integer(5),Integer(3)); S = ZZ['x']; (x,) = S._first_ngens(1); f = x**Integer(5) + Integer(25)*x**Integer(2) - Integer(5); W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1)
>>> W.random_element().parent() is W
True


Bases: Map

A base class for various isomorphisms between p-adic rings/fields and free modules

EXAMPLES:

sage: K.<a> = Qq(125)
sage: V, fr, to = K.free_module()
True

>>> from sage.all import *
>>> K = Qq(Integer(125), names=('a',)); (a,) = K._first_ngens(1)
>>> V, fr, to = K.free_module()
True

is_injective()[source]#

EXAMPLES:

sage: K.<a> = Qq(125)
sage: V, fr, to = K.free_module()
sage: fr.is_injective()
True

>>> from sage.all import *
>>> K = Qq(Integer(125), names=('a',)); (a,) = K._first_ngens(1)
>>> V, fr, to = K.free_module()
>>> fr.is_injective()
True

is_surjective()[source]#

EXAMPLES:

sage: K.<a> = Qq(125)
sage: V, fr, to = K.free_module()
sage: fr.is_surjective()
True

>>> from sage.all import *
>>> K = Qq(Integer(125), names=('a',)); (a,) = K._first_ngens(1)
>>> V, fr, to = K.free_module()
>>> fr.is_surjective()
True