Elements in a Henselization which are described by a limit valuation¶

AUTHORS:

Julian Rüth (2016-11-16): initial version

class henselization.sage.rings.padics.henselization.mac_lane_element.MacLaneElement_Field(parent, limit_valuation, degree)¶

Bases: henselization.sage.rings.padics.henselization.mac_lane_element.MacLaneElement_base, henselization.sage.rings.padics.henselization.henselization_element.HenselizationElement_Field

A MacLaneElement_base that lives in a field.

EXAMPLES:

sage: from henselization import *
sage: R.<x> = QQ.henselization(5)[]
sage: F = (x^2 + 1).factor()
sage: a = F[0][0][0]; a
2 + O(5^10)

TESTS:

sage: from sage.rings.padics.henselization.mac_lane_element import MacLaneElement_Field
sage: isinstance(a, MacLaneElement_Field)
True
sage: TestSuite(a).run() # long time

class henselization.sage.rings.padics.henselization.mac_lane_element.MacLaneElement_Ring(parent, limit_valuation, degree)¶

Bases: henselization.sage.rings.padics.henselization.mac_lane_element.MacLaneElement_base, henselization.sage.rings.padics.henselization.henselization_element.HenselizationElement_Ring

A MacLaneElement_base that lives in a ring that is not a field.

EXAMPLES:

sage: from henselization import *
sage: S = ZZ.henselization(5)
sage: R.<x> = S[]
sage: F = (x^2 + 1).factor()
sage: a = F[0][0][0]; a
2 + O(5)

TESTS:

sage: from sage.rings.padics.henselization.mac_lane_element import MacLaneElement_Ring
sage: isinstance(a, MacLaneElement_Ring)
True
sage: TestSuite(a).run() # long time

class henselization.sage.rings.padics.henselization.mac_lane_element.MacLaneElement_base(parent, limit_valuation, degree)¶

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

Element class for elements of CompleteRing_base which are given by the limit of the coefficients in a specific degree of the key polynomials of a MacLaneLimitValuation.

EXAMPLES:

sage: from henselization import *
sage: K = QQ.henselization(5)
sage: R.<x> = K[]
sage: F = (x^2 + 1).factor()

The coefficients of the factors are the limits of the coefficients of the key polynomials:

sage: F
(x + 2 + O(5)) * (x + 3 + O(5))

approximation(precision)¶

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

EXAMPLES:

sage: from henselization import *
sage: K = QQ.henselization(5)
sage: R.<x> = K[]
sage: a = (x^2 + 1).factor()[0][0][0]
sage: a
2 + O(5)
sage: a.approximation(precision=10)
6139557

reduction()¶

Return the reduction of this element module the element of positive valuation().

EXAMPLES:

sage: from henselization import *
sage: K = QQ.henselization(5)
sage: R.<x> = K[]
sage: a = (x^2 + 1).factor()[0][0][0]
sage: a.reduction()
2

valuation()¶

Return the valuation of this element.

EXAMPLES:

sage: from henselization import *
sage: K = QQ.henselization(5)
sage: R.<x> = K[]
sage: a = (x^2 + 1).factor()[0][0][0]
sage: a.valuation()
0