Unramified Extension Generic¶
This file implements the shared functionality for unramified extensions.
AUTHORS:
- David Roe
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class
sage.rings.padics.unramified_extension_generic.
UnramifiedExtensionGeneric
(poly, prec, print_mode, names, element_class)¶ Bases:
sage.rings.padics.padic_extension_generic.pAdicExtensionGeneric
An unramified extension of Qp or Zp.
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discriminant
(K=None)¶ Returns the discriminant of self over the subring K.
INPUT:
- K – a subring/subfield (defaults to the base ring).
EXAMPLES:
sage: R.<a> = Zq(125) sage: R.discriminant() Traceback (most recent call last): ... NotImplementedError
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gen
(n=0)¶ Returns 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() a + O(5^20)
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has_pth_root
()¶ Returns whether or not \(\ZZ_p\) has a primitive \(p^{\mbox{th}}\) root of unity.
Since adjoining a \(p^{\mbox{th}}\) 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 \(p^{\mbox{th}}\) root of unity.
EXAMPLES:
sage: R.<a> = Zq(1024); R.has_pth_root() True sage: R.<a> = Zq(17^5); R.has_pth_root() False
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has_root_of_unity
(n)¶ Returns whether or not \(\ZZ_p\) has a primitive \(n^{\mbox{th}}\) root of unity.
INPUT:
- self – a p-adic ring
- n – an integer
OUTPUT:
- boolean – whether self has primitive \(n^{\mbox{th}}\) root of unity
EXAMPLES:
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
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inertia_degree
(K=None)¶ Returns the inertia degree of self over the subring K.
INPUT:
- K – a subring (or subfield) of self. Defaults to the base.
EXAMPLES:
sage: R.<a> = Zq(125); R.inertia_degree() 3
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is_galois
(K=None)¶ Returns 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() True
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ramification_index
(K=None)¶ Returns the ramification index of self over the subring K.
INPUT:
- K – a subring (or subfield) of self. Defaults to the base.
EXAMPLES:
sage: R.<a> = Zq(125); R.ramification_index() 1
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residue_class_field
()¶ Returns the residue class field.
EXAMPLES:
sage: R.<a> = Zq(125); R.residue_class_field() Finite Field in a0 of size 5^3
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uniformizer
()¶ Returns 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) sage: R.uniformizer() 5 + O(5^21)
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uniformizer_pow
(n)¶ Returns the nth power of the uniformizer of self (as an element of self).
EXAMPLES:
sage: R.<a> = ZqCR(125) sage: R.uniformizer_pow(5) 5^5 + O(5^25)
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