Symbolic elements which generate an extension of a Henselization

AUTHORS:

• Julian Rüth (2017-04-30): initial version

class henselization.sage.rings.padics.henselization.generator_element.GeneratorElement(parent)

Bases: henselization.sage.rings.padics.henselization.henselization_element.HenselizationElement_base

Element class for generators of HenselizationExtension_base which merely exist symbolically as the root of a certain polynomial but which support essentially no arithmetic.

EXAMPLES:

sage: from henselization import *
sage: K = QQ.henselization(3)
sage: R.<x> = K[]
sage: L = K.extension(x^2 - 3)
sage: R.<y> = L[]
sage: M = L.extension(y^3 - 3)
sage: M.gen()
y

In towers of extensions, these elements are also used to make explicit the choice of embedding (however, this is not implemented yet):

sage: M(L.gen()) == -M(L.gen())
Traceback (most recent call last):
...
NotImplementedError: Selection of approximate root of x^2 - 3 in Extension defined by y^3 - 3 of Extension defined by x^2 - 3 of Henselization of Rational Field with respect to 3-adic valuation