This file implements the shared functionality for Eisenstein extensions.
AUTHORS:
Bases: sage.rings.padics.padic_extension_generic.pAdicExtensionGeneric
Initializes self.
EXAMPLES:
sage: A = Zp(7,10)
sage: S.<x> = A[]
sage: B.<t> = A.ext(x^2+7) #indirect doctest
Returns a generator for self as an extension of its ground ring.
EXAMPLES:
sage: A = Zp(7,10)
sage: S.<x> = A[]
sage: B.<t> = A.ext(x^2+7)
sage: B.gen()
t + O(t^21)
Returns the inertia degree of self over K, or the ground ring if K is None.
The inertia degree is the degree of the extension of residue fields induced by this extensions. Since Eisenstein extensions are totally ramified, this will be 1 for K=None.
INPUTS:
OUTPUTS:
EXAMPLES:
sage: A = Zp(7,10)
sage: S.<x> = A[]
sage: B.<t> = A.ext(x^2+7)
sage: B.inertia_degree()
1
Returns the inertia subring.
Since an Eisenstein extension is totally ramified, this is just the ground field.
EXAMPLES:
sage: A = Zp(7,10)
sage: S.<x> = A[]
sage: B.<t> = A.ext(x^2+7)
sage: B.inertia_subring()
7-adic Ring with capped relative precision 10
Returns the ramification index of self over K, or over the ground ring if K is None.
The ramification index is the index of the image of the valuation map on K in the image of the valuation map on self (both normalized so that the valuation of p is 1).
INPUTS:
OUTPUTS:
EXAMPLES:
sage: A = Zp(7,10)
sage: S.<x> = A[]
sage: B.<t> = A.ext(x^2+7)
sage: B.ramification_index()
2
Returns the residue class field.
INPUT:
OUTPUT:
EXAMPLES:
sage: A = Zp(7,10)
sage: S.<x> = A[]
sage: B.<t> = A.ext(x^2+7)
sage: B.residue_class_field()
Finite Field of size 7
Returns the uniformizer of self, ie a generator for the unique maximal ideal.
EXAMPLES:
sage: A = Zp(7,10)
sage: S.<x> = A[]
sage: B.<t> = A.ext(x^2+7)
sage: B.uniformizer()
t + O(t^21)
Returns the nth power of the uniformizer of self (as an element of self).
EXAMPLES:
sage: A = Zp(7,10)
sage: S.<x> = A[]
sage: B.<t> = A.ext(x^2+7)
sage: B.uniformizer_pow(5)
t^5 + O(t^25)