Unramified Extension Generic#

This file implements the shared functionality for unramified extensions.

AUTHORS:

David Roe

class sage.rings.padics.unramified_extension_generic.UnramifiedExtensionGeneric(poly, prec, print_mode, names, element_class)#

Bases: pAdicExtensionGeneric

An unramified extension of $Q_{p}$ or $Z_{p}$ .

absolute_f()#

Return the degree of the residue field of this ring/field over its prime subfield.

EXAMPLES:

sage: K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
sage: K.absolute_f()                                                        # needs sage.libs.ntl
5

sage: x = polygen(ZZ, 'x')
sage: L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
sage: L.absolute_f()                                                        # needs sage.libs.ntl
1

K.<a> = Qq(3^5)                                                       # needs sage.libs.ntl
K.absolute_f()                                                        # needs sage.libs.ntl
x = polygen(ZZ, 'x')
L.<pi> = Qp(3).extension(x^2 - 3)                                     # needs sage.libs.ntl
L.absolute_f()                                                        # needs sage.libs.ntl

discriminant(K=None)#

Return the discriminant of self over the subring $K$ .

INPUT:

K – a subring/subfield (defaults to the base ring).

EXAMPLES:

sage: R.<a> = Zq(125)                                                       # needs sage.libs.ntl
sage: R.discriminant()                                                      # needs sage.libs.ntl
Traceback (most recent call last):
...
NotImplementedError

R.<a> = Zq(125)                                                       # needs sage.libs.ntl
R.discriminant()                                                      # needs sage.libs.ntl

gen(n=0)#

Return a generator for this unramified extension.

This is an element that satisfies the polynomial defining this extension. Such an element will reduce to a generator of the corresponding residue field extension.

EXAMPLES:

sage: R.<a> = Zq(125); R.gen()                                              # needs sage.libs.ntl
a + O(5^20)

R.<a> = Zq(125); R.gen()                                              # needs sage.libs.ntl

has_pth_root()#

Return whether or not $Z_{p}$ has a primitive $p$ -th root of unity.

Since adjoining a $p$ -th root of unity yields a totally ramified extension, self will contain one if and only if the ground ring does.

INPUT:

self – a $p$ -adic ring

OUTPUT:

boolean – whether self has primitive $p$ -th root of unity.

EXAMPLES:

sage: R.<a> = Zq(1024); R.has_pth_root()                                    # needs sage.libs.ntl
True
sage: R.<a> = Zq(17^5); R.has_pth_root()                                    # needs sage.libs.ntl
False

R.<a> = Zq(1024); R.has_pth_root()                                    # needs sage.libs.ntl
R.<a> = Zq(17^5); R.has_pth_root()                                    # needs sage.libs.ntl

has_root_of_unity(n)#

Return whether or not $Z_{p}$ has a primitive $n$ -th root of unity.

INPUT:

self – a $p$ -adic ring
n – an integer

OUTPUT:

boolean

EXAMPLES:

sage: # needs sage.libs.ntl
sage: R.<a> = Zq(37^8)
sage: R.has_root_of_unity(144)
True
sage: R.has_root_of_unity(89)
True
sage: R.has_root_of_unity(11)
False

# needs sage.libs.ntl
R.<a> = Zq(37^8)
R.has_root_of_unity(144)
R.has_root_of_unity(89)
R.has_root_of_unity(11)

is_galois(K=None)#

Return True if this extension is Galois.

Every unramified extension is Galois.

INPUT:

K – a subring/subfield (defaults to the base ring).

EXAMPLES:

sage: R.<a> = Zq(125); R.is_galois()                                        # needs sage.libs.ntl
True

R.<a> = Zq(125); R.is_galois()                                        # needs sage.libs.ntl

residue_class_field()#

Returns the residue class field.

EXAMPLES:

sage: R.<a> = Zq(125); R.residue_class_field()                              # needs sage.libs.ntl
Finite Field in a0 of size 5^3

R.<a> = Zq(125); R.residue_class_field()                              # needs sage.libs.ntl

residue_ring(n)#

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

EXAMPLES:

sage: R.<a> = Zq(125)                                                       # needs sage.libs.ntl
sage: R.residue_ring(1)                                                     # needs sage.libs.ntl
Finite Field in a0 of size 5^3

R.<a> = Zq(125)                                                       # needs sage.libs.ntl
R.residue_ring(1)                                                     # needs sage.libs.ntl

The following requires implementing more general Artinian rings:

sage: R.residue_ring(2)                                                     # needs sage.libs.ntl
Traceback (most recent call last):
...
NotImplementedError

R.residue_ring(2)                                                     # needs sage.libs.ntl

uniformizer()#

Return a uniformizer for this extension.

Since this extension is unramified, a uniformizer for the ground ring will also be a uniformizer for this extension.

EXAMPLES:

sage: R.<a> = ZqCR(125)                                                     # needs sage.libs.ntl
sage: R.uniformizer()                                                       # needs sage.libs.ntl
5 + O(5^21)

R.<a> = ZqCR(125)                                                     # needs sage.libs.ntl
R.uniformizer()                                                       # needs sage.libs.ntl

uniformizer_pow(n)#

Return the $n$ -th power of the uniformizer of self (as an element of self).

EXAMPLES:

sage: R.<a> = ZqCR(125)                                                     # needs sage.libs.ntl
sage: R.uniformizer_pow(5)                                                  # needs sage.libs.ntl
5^5 + O(5^25)

R.<a> = ZqCR(125)                                                     # needs sage.libs.ntl
R.uniformizer_pow(5)                                                  # needs sage.libs.ntl