Eisenstein Extension Generic#
This file implements the shared functionality for Eisenstein extensions.
AUTHORS:
David Roe
- class sage.rings.padics.eisenstein_extension_generic.EisensteinExtensionGeneric(poly, prec, print_mode, names, element_class)#
Bases:
pAdicExtensionGenericInitializes
self.EXAMPLES:
sage: A = Zp(7,10) sage: S.<x> = A[] # needs sage.libs.ntl sage: B.<t> = A.ext(x^2+7) # indirect doctest # needs sage.libs.ntl sage.rings.padics
- absolute_e()#
Return the absolute ramification index of this ring or field
EXAMPLES:
sage: K.<a> = Qq(3^5) # needs sage.libs.ntl sage: K.absolute_e() # needs sage.libs.ntl 1 sage: x = polygen(ZZ, 'x') sage: L.<pi> = Qp(3).extension(x^2 - 3) # needs sage.libs.ntl sage: L.absolute_e() # needs sage.libs.ntl 2
- gen(n=0)#
Return a generator for
selfas an extension of its ground ring.EXAMPLES:
sage: A = Zp(7,10) sage: S.<x> = A[] # needs sage.libs.ntl sage: B.<t> = A.ext(x^2 + 7) # needs sage.libs.ntl sage: B.gen() # needs sage.libs.ntl t + O(t^21)
- inertia_subring()#
Return 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[] # needs sage.libs.ntl sage: B.<t> = A.ext(x^2 + 7) # needs sage.libs.ntl sage: B.inertia_subring() # needs sage.libs.ntl 7-adic Ring with capped relative precision 10
- residue_class_field()#
Return the residue class field.
INPUT:
self– a p-adic ring
OUTPUT:
the residue field
EXAMPLES:
sage: A = Zp(7,10) sage: S.<x> = A[] # needs sage.libs.ntl sage: B.<t> = A.ext(x^2 + 7) # needs sage.libs.ntl sage: B.residue_class_field() # needs sage.libs.ntl Finite Field of size 7
- residue_ring(n)#
Return the quotient of the ring of integers by the \(n\)-th power of its maximal ideal.
EXAMPLES:
sage: S.<x> = ZZ[] sage: W.<w> = Zp(5).extension(x^2 - 5) # needs sage.libs.ntl sage: W.residue_ring(1) # needs sage.libs.ntl Ring of integers modulo 5
The following requires implementing more general Artinian rings:
sage: W.residue_ring(2) # needs sage.libs.ntl Traceback (most recent call last): ... NotImplementedError
- uniformizer()#
Return the uniformizer of
self, i.e., a generator for the unique maximal ideal.EXAMPLES:
sage: A = Zp(7,10) sage: S.<x> = A[] # needs sage.libs.ntl sage: B.<t> = A.ext(x^2 + 7) # needs sage.libs.ntl sage: B.uniformizer() # needs sage.libs.ntl t + O(t^21)
- uniformizer_pow(n)#
Return the \(n\)-th power of the uniformizer of
self(as an element ofself).EXAMPLES:
sage: A = Zp(7,10) sage: S.<x> = A[] # needs sage.libs.ntl sage: B.<t> = A.ext(x^2 + 7) # needs sage.libs.ntl sage: B.uniformizer_pow(5) # needs sage.libs.ntl t^5 + O(t^25)