Elements of the Henselization that come from the uncompleted ring¶

AUTHORS:

Julian Rüth (2016-11-15): initial version

class henselization.sage.rings.padics.henselization.base_element.BaseElement_Field(parent, base, valuation, x)¶

Bases: henselization.sage.rings.padics.henselization.base_element.BaseElement_base, henselization.sage.rings.padics.henselization.henselization_element.HenselizationElement_Field

An element of Henselization_Field which is in one of its uncompleted base fields.

EXAMPLES:

sage: from henselization import *
sage: K = QQ.henselization(2)
sage: K(0)
0

class henselization.sage.rings.padics.henselization.base_element.BaseElement_Ring(parent, base, valuation, x)¶

Bases: henselization.sage.rings.padics.henselization.base_element.BaseElement_base, henselization.sage.rings.padics.henselization.henselization_element.HenselizationElement_Ring

An element of Henselization_Ring which is in one of its uncompleted base fields.

EXAMPLES:

sage: from henselization import *
sage: R = ZZ.henselization(2)
sage: x = R(0); x
0

TESTS:

sage: from sage.rings.padics.henselization.base_element import BaseElement_Ring
sage: isinstance(R(0), BaseElement_Ring)
True

class henselization.sage.rings.padics.henselization.base_element.BaseElement_base(parent, base, valuation, x)¶

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

Abstract base class for elements of Henselization_base which are in one of its uncompleted base fields.

EXAMPLES:

sage: from henselization import *
sage: K = QQ.henselization(2)
sage: x = K(0); x
0

approximation(precision=None)¶

Return an approximation to this element which is know to differ from the actual element by at most precision.

EXAMPLES:

sage: from henselization import *
sage: K = QQ.henselization(2)
sage: R.<x> = K[]
sage: L = K.extension(x^2 + x + 1)
sage: L.gen().approximation(123)
x

matrix(base=None)¶

Return the matrix of this element over base.

EXAMPLES:

sage: from henselization import *
sage: K = QQ.henselization(2)
sage: R.<x> = K[]
sage: L = K.extension(x^2 + x + 1)
sage: L.gen().matrix(base=K)
[ 0  1]
[-1 -1]

reduction()¶

Return the reduction of this element modulo the elements of positive valuation().

EXAMPLES:

sage: from henselization import *
sage: K = QQ.henselization(2)
sage: x = K(4)
sage: x.reduction()
0

simplify(error=None, force=False)¶

Return a simplified version of this element.

Produce an element which differs from this element by an element of valuation strictly greater than the valuation of this element (or strictly greater than error if set.)

EXAMPLES:

sage: from henselization import *
sage: K = QQ.henselization(2)
sage: K(1025).simplify(force=True)
1