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)

An unramified extension of Qp or Zp.

__init__(poly, prec, print_mode, names, element_class)

Initializes self

INPUTS:

- poly -- Polynomial defining this extension.
- prec -- The precision cap
- print_mode -- a dictionary with print options            
- names -- a 4-tuple, (variable_name, residue_name,
  unramified_subextension_variable_name, uniformizer_name)
- element_class -- the class for elements of this unramified extension.

EXAMPLES:

sage: R.<a> = Zq(27) #indirect doctest
_repr_(do_latex=False)

Representation.

EXAMPLES:

sage: R.<a> = Zq(125); R #indirect doctest
Unramified Extension of 5-adic Ring with capped absolute precision 20 in a defined by (1 + O(5^20))*x^3 + (3 + O(5^20))*x + (3 + O(5^20))
sage: latex(R) #indirect doctest
\mathbf{Z}_{5^{3}}
_uniformizer_print()

Returns how the uniformizer is supposed to print.

EXAMPLES:

sage: R.<a> = Zq(125); R._uniformizer_print()
'5'
_unram_print()

Returns how the generator prints.

EXAMPLES:

sage: R.<a> = Zq(125); R._unram_print()
'a'
discriminant(K=None)

Returns the discriminant of self over the subring K.

INPUTS:

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

EXAMPLES:

sage: R.<a> = Zq(125)
sage: R.discriminant()
...
NotImplementedError
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)
has_pth_root()

Returns whether or not $Z_p$ 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
has_root_of_unity(n)

Returns whether or not $Z_p$ 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
inertia_degree(K=None)

Returns the inertia degree of self over the subring K.

INPUTS:

- K -- a subring (or subfield) of self.  Defaults to the
  base.

EXAMPLES:

sage: R.<a> = Zq(125); R.inertia_degree()
3
is_galois(K=None)

Returns True if this extension is Galois.

Every unramified extension is Galois.

INPUTS:

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

EXAMPLES:

sage: R.<a> = Zq(125); R.is_galois()
True
ramification_index(K=None)

Returns the ramification index of self over the subring K.

INPUTS:

- K -- a subring (or subfield) of self.  Defaults to the
  base.

EXAMPLES:

sage: R.<a> = Zq(125); R.ramification_index()
1
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
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)
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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