Eisenstein Extension Generic¶
This file implements the shared functionality for Eisenstein extensions.
AUTHORS:
- David Roe
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class
sage.rings.padics.eisenstein_extension_generic.
EisensteinExtensionGeneric
(poly, prec, print_mode, names, element_class)¶ 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
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gen
(n=0)¶ 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)
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inertia_degree
(K=None)¶ 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.
INPUT:
- self – an Eisenstein extension
- K – a subring of self (default None -> self.ground_ring())
OUTPUT:
- The degree of the induced extensions of residue fields.
EXAMPLES:
sage: A = Zp(7,10) sage: S.<x> = A[] sage: B.<t> = A.ext(x^2+7) sage: B.inertia_degree() 1
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inertia_subring
()¶ 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
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ramification_index
(K=None)¶ 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).
INPUT:
- self – an Eisenstein extension
- K – a subring of self (default None -> self.ground_ring())
OUTPUT:
- The ramification index of the extension self/K
EXAMPLES:
sage: A = Zp(7,10) sage: S.<x> = A[] sage: B.<t> = A.ext(x^2+7) sage: B.ramification_index() 2
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residue_class_field
()¶ Returns the residue class field.
INPUT:
- self – a p-adic ring
OUTPUT:
- the residue field
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
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uniformizer
()¶ 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)
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uniformizer_pow
(n)¶ 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)
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