Helper classes for structural embeddings and isomorphisms of number fields¶
AUTHORS:
- Julian Rueth (2014-04-03): initial version
Consider the following fields \(L\) and \(M\):
sage: L.<a> = QuadraticField(2)
sage: M.<a> = L.absolute_field()
Both produce the same extension of \(\QQ\). However, they should not be identical because \(M\) carries additional information:
sage: L.structure()
(Ring Coercion endomorphism of Number Field in a with defining polynomial x^2 - 2,
Ring Coercion endomorphism of Number Field in a with defining polynomial x^2 - 2)
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,
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)
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
-
class
sage.rings.number_field.structure.
AbsoluteFromRelative
(other)¶ Bases:
sage.rings.number_field.structure.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.
TESTS:
sage: from sage.rings.number_field.structure import AbsoluteFromRelative sage: K.<a> = QuadraticField(2) sage: R.<x> = K[] sage: L.<b> = K.extension(x^2 - 3) sage: AbsoluteFromRelative(L) <sage.rings.number_field.structure.AbsoluteFromRelative object at 0x...>
-
create_structure
(field)¶ Return a pair of isomorphisms which go from
field
toother
and vice versa.TESTS:
sage: K.<a> = QuadraticField(2) sage: R.<x> = K[] sage: L.<b> = K.extension(x^2 - 3) sage: M.<c> = L.absolute_field() sage: M.structure() # indirect doctest (Isomorphism map: From: Number Field in c with defining polynomial x^4 - 10*x^2 + 1 To: Number Field in b with defining polynomial x^2 - 3 over its base field, Isomorphism map: From: Number Field in b with defining polynomial x^2 - 3 over its base field To: Number Field in c with defining polynomial x^4 - 10*x^2 + 1)
-
class
sage.rings.number_field.structure.
NameChange
(other)¶ Bases:
sage.rings.number_field.structure.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.
TESTS:
sage: from sage.rings.number_field.structure import NameChange sage: K.<i> = QuadraticField(-1) sage: NameChange(K) <sage.rings.number_field.structure.NameChange object at 0x...>
-
create_structure
(field)¶ Return a pair of isomorphisms which send the generator of
field
to the generator ofother
and vice versa.TESTS:
sage: CyclotomicField(5).absolute_field('a').structure() # indirect doctest (Isomorphism given by variable name change map: From: Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1 To: Cyclotomic Field of order 5 and degree 4, Isomorphism given by variable name change map: From: Cyclotomic Field of order 5 and degree 4 To: Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1)
-
class
sage.rings.number_field.structure.
NumberFieldStructure
(other)¶ Bases:
sage.structure.unique_representation.UniqueRepresentation
Abstract base class encapsulating information about a number fields relation to other number fields.
TESTS:
sage: from sage.rings.number_field.structure import NumberFieldStructure sage: NumberFieldStructure(QQ) <sage.rings.number_field.structure.NumberFieldStructure object at 0x...>
Instances are cached through
sage.structure.unique_representation.UniqueRepresentation
:sage: NumberFieldStructure(QQ) is NumberFieldStructure(QQ) True sage: R.<x> = QQ[] sage: K.<i> = NumberField(x^2+1) sage: L = K.change_names('j').change_names('i') sage: K is L # K and L differ in "structure", one is the "name-change" of the other False sage: NumberFieldStructure(L) is NumberFieldStructure(L) True sage: NumberFieldStructure(K) is NumberFieldStructure(L) False sage: from sage.rings.number_field.structure import NameChange sage: KK.<j> = NumberField(x^2+1, structure=NameChange(K)) sage: LL.<j> = NumberField(x^2+1, structure=NameChange(L)) sage: KK is LL False
-
create_structure
(field)¶ 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 whichfield
has been constructed and the second one its inverse. In this case, these morphisms are used as conversion maps between the two fields.TESTS:
sage: from sage.rings.number_field.structure import NumberFieldStructure sage: NumberFieldStructure(QQ).create_structure(QQ) Traceback (most recent call last): ... NotImplementedError
The morphisms created by this method are used as conversion maps:
sage: K.<i> = QuadraticField(-1) sage: L.<j> = K.change_names() sage: isinstance(L._structure, NumberFieldStructure) True sage: from_L, to_L = L.structure() sage: L._convert_map_from_(K) is to_L True sage: L(i) j sage: K(j) i
-
-
class
sage.rings.number_field.structure.
RelativeFromAbsolute
(other, gen)¶ Bases:
sage.rings.number_field.structure.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
TESTS:
sage: from sage.rings.number_field.structure import RelativeFromAbsolute sage: RelativeFromAbsolute(QQ, 1/2) <sage.rings.number_field.structure.RelativeFromAbsolute object at 0x...>
-
create_structure
(field)¶ Return a pair of isomorphisms which go from
field
toother
and vice versa.INPUT:
field
– a relative number field
TESTS:
sage: K.<a> = QuadraticField(2) sage: M.<b,a_> = K.relativize(-a) sage: M.structure() # indirect doctest (Relative number field morphism: From: Number Field in b with defining polynomial x + a_ over its base field To: Number Field in a with defining polynomial x^2 - 2 Defn: -a_ |--> a a_ |--> -a, Ring morphism: From: Number Field in a with defining polynomial x^2 - 2 To: Number Field in b with defining polynomial x + a_ over its base field Defn: a |--> -a_)
-
class
sage.rings.number_field.structure.
RelativeFromRelative
(other)¶ Bases:
sage.rings.number_field.structure.NumberFieldStructure
Structure for a relative number field created from another relative number field.
INPUT:
other
– the relative number field used in the construction, seecreate_structure()
; there this field will be calledfield_
.
TESTS:
sage: from sage.rings.number_field.structure import RelativeFromRelative sage: K.<i> = QuadraticField(-1) sage: R.<x> = K[] sage: L.<a> = K.extension(x^2 - 2) sage: RelativeFromRelative(L) <sage.rings.number_field.structure.RelativeFromRelative object at 0x...>
-
create_structure
(field)¶ Return a pair of isomorphisms which go from
field
to the relative number field (calledother
below) from whichfield
has been created and vice versa.The isomorphism is created via the relative number field
field_
which is identical tofield
but is equipped with an isomorphism to an absolute field which was used in the construction offield
.INPUT:
field
– a relative number field
TESTS:
sage: K.<i> = QuadraticField(-1) sage: R.<x> = K[] sage: L.<a> = K.extension(x^2 - 2) sage: M.<b,a> = L.relativize(a) sage: M.structure() # indirect doctest (Relative number field morphism: From: Number Field in b with defining polynomial x^2 - 2*a*x + 3 over its base field To: Number Field in a with defining polynomial x^2 - 2 over its base field Defn: b |--> a - i a |--> a, Relative number field morphism: From: Number Field in a with defining polynomial x^2 - 2 over its base field To: Number Field in b with defining polynomial x^2 - 2*a*x + 3 over its base field Defn: a |--> a i |--> -b + a)