Helper classes for structural embeddings and isomorphisms of number fields

Consider the following fields \(L\) and \(M\):

sage: L.<a> = QuadraticField(2)
sage: M.<a> = L.absolute_field()

>>> from sage.all import *
>>> L = QuadraticField(Integer(2), names=('a',)); (a,) = L._first_ngens(1)
>>> M = L.absolute_field(names=('a',)); (a,) = M._first_ngens(1)

Both produce the same extension of \(\QQ\). However, they should not be identical because \(M\) carries additional information:

sage: L.structure()
(Identity endomorphism of
  Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095?,
 Identity endomorphism of
  Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095?)
sage: M.structure()
(Isomorphism given by variable name change map:
   From: Number Field in a with defining polynomial x^2 - 2
   To:   Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095?,
 Isomorphism given by variable name change map:
   From: Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095?
   To:   Number Field in a with defining polynomial x^2 - 2)

>>> from sage.all import *
>>> L.structure()
(Identity endomorphism of
  Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095?,
 Identity endomorphism of
  Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095?)
>>> M.structure()
(Isomorphism given by variable name change map:
   From: Number Field in a with defining polynomial x^2 - 2
   To:   Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095?,
 Isomorphism given by variable name change map:
   From: Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095?
   To:   Number Field in a with defining polynomial x^2 - 2)

This used to cause trouble with caching and made (absolute) number fields not unique when they should have been. The underlying technical problem is that the morphisms returned by structure() can only be defined once the fields in question have been created. Therefore, these morphisms cannot be part of a key which uniquely identifies a number field.

The classes defined in this file encapsulate information about these structure morphisms which can be passed to the factory creating number fields. This makes it possible to distinguish number fields which only differ in terms of these structure morphisms:

sage: L is M
False
sage: N.<a> = L.absolute_field()
sage: M is N
True

>>> from sage.all import *
>>> L is M
False
>>> N = L.absolute_field(names=('a',)); (a,) = N._first_ngens(1)
>>> M is N
True

AUTHORS:

  • Julian Rueth (2014-04-03): initial version

class sage.rings.number_field.structure.AbsoluteFromRelative(other)[source]

Bases: NumberFieldStructure

Structure for an absolute number field created from a relative number field.

INPUT:

  • other – the number field from which this field has been created

create_structure(field)[source]

Return a pair of isomorphisms which go from field to other and vice versa.

class sage.rings.number_field.structure.NameChange(other)[source]

Bases: NumberFieldStructure

Structure for a number field created by a change in variable name.

INPUT:

  • other – the number field from which this field has been created

create_structure(field)[source]

Return a pair of isomorphisms which send the generator of field to the generator of other and vice versa.

class sage.rings.number_field.structure.NumberFieldStructure(other)[source]

Bases: UniqueRepresentation

Abstract base class encapsulating information about a number fields relation to other number fields.

create_structure(field)[source]

Return a tuple encoding structural information about field.

OUTPUT:

Typically, the output is a pair of morphisms. The first one from field to a field from which field has been constructed and the second one its inverse. In this case, these morphisms are used as conversion maps between the two fields.

class sage.rings.number_field.structure.RelativeFromAbsolute(other, gen)[source]

Bases: NumberFieldStructure

Structure for a relative number field created from an absolute number field.

INPUT:

  • other – the (absolute) number field from which this field has been created

  • gen – the generator of the intermediate field

create_structure(field)[source]

Return a pair of isomorphisms which go from field to other and vice versa.

INPUT:

  • field – a relative number field

class sage.rings.number_field.structure.RelativeFromRelative(other)[source]

Bases: NumberFieldStructure

Structure for a relative number field created from another relative number field.

INPUT:

  • other – the relative number field used in the construction, see create_structure(); there this field will be called field_

create_structure(field)[source]

Return a pair of isomorphisms which go from field to the relative number field (called other below) from which field has been created and vice versa.

The isomorphism is created via the relative number field field_ which is identical to field but is equipped with an isomorphism to an absolute field which was used in the construction of field.

INPUT:

  • field – a relative number field