Fields with a Labelled Embedding#

A Number Field has several diffenent embeddings into the complex numbers. An embedded Number Field is a Number Field such that one of these embeddings is labelled. These fields form a category named FldNumEmb.

Creation Functions#

Functions are provided to create fields of the special type FldNumEmb.

EmbeddedNumberField(L, c): FldNum, FldComElt -> BoolElt, FldNumEmb#
EmbeddedNumberField(L, c): FldNum, FldReElt -> BoolElt, FldNumEmb#: Attempts to turn \(L\) into a Number Field with labelled embedding, such that its generator is mapped to a complex number with approximation \(c\). It returns true if such an embedding can be identified, otherwise false.

EmbeddedNumberField(f, r): RngUPolElt, FldReElt -> BoolElt, FldNumEmb#
EmbeddedNumberField(f, r): RngUPolElt, FldComElt -> BoolElt, FldNumEmb#: Attempts to construct a Number Field with defining polynomial \(f\) and a labelled embedding, such that its generator is mapped to a complex number with approximation \(r\). It returns true if such an embedding can be identified, otherwise false.

EmbeddedNumberField(f, i): RngUPolElt, RngIntElt -> FldNumEmb#: Constructs a Number Field with defining polynomial \(f\) and labelled embedding with number \(i\).

EmbeddedNumberField(L, i): FldNum, RngIntElt -> FldNumEmb#: Constructs a Number Field with labelled embedding with number \(i\) and the same defining polynomial as \(L\).

EmbeddedSplittingField(f): RngUPolElt -> FldNumEmb, SeqEnum#: Constructs the splitting field of \(f\) as number field with labelled embedding. The second return value are the roots of \(f\) in the splitting field.

Subfield(L, K): FldNumEmb, FldNum -> FldNumEmb#: Turns the subfield \(K\) of the Number Field \(L\) with labelled embedding into a number field with labelled embedding.

Composition and Intersection#

Number fields with a fixed embedding into the complex numbers have a unique composition and intersection as subfields of the complex numbers. The following functions are availabe to work in the subfield lattice.

Composite(K, L): FldNumEmb, FldNumEmb -> FldNumEmb#: The unique composite of \(K\) and \(L\) as a number field with labelled embedding.

Intersection(K, L): FldNumEmb, FldNumEmb -> FldNumEmb#
K meet L: FldNumEmb, FldNumEmb -> FldNumEmb#: The unique intersection of \(K\) and \(L\) as a number field with labelled embedding.

IsSubfieldEmb(K, L): FldNumEmb, FldNumEmb -> BoolElt#: Returns true if the image of \(K\) in the complex numbers is a subfield of the image of \(L\) and false otherwise.

Embedding and Reconstruction#

The following functions are available to work with elements of a number field with a labelled embedding.

ComplexImage(x): FldNumElt -> FldComElt#

Precision: RngIntElt                    Default: 30

The image of \(x\) in the complex numbers.

Embedding(K): FldNumEmb -> UserProgram#

Precision: RngIntElt                    Default: 30

The labelled embeding of \(K\) into the complex numbers as a user program.

Reconstruction(L, x): FldNumEmb, FldComElt -> BoolElt, FldNumElt#

Reconstruction(L, x): FldNumEmb, FldReElt -> BoolElt, FldNumElt#

UseLLLOrder: BoolElt                    Default: false

Attempts to reconstruct \(x\) as an element in \(L\). Returns true if successful and false otherwise. If the option UseLLLOrder is set, and LLL basis of the maximal order will be used.

Example: Fldemb Eg 1 (ex-83530f)#

Here is a basic example:

> _<x> := PolynomialRing(Integers());
> K := NumberField(x^3-7);
> suc, L := EmbeddedNumberField(K,7^(1/3));
> ComplexImage(L.1 + L.1^2);
5.57223689279536061843719014967
> Reconstruction(L,1/2 + 7^(1/3) + 5 * 7^(2/3));
true 1/2*(10*L.1^2 + 2*L.1 + 1)
> Reconstruction(L,Sqrt(-1));
false

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Intersection and composition can be used as follows:

> _<x> := PolynomialRing(Integers());
> suc, K := EmbeddedNumberField(x^6-3,3^(1/6));
> suc, L := EmbeddedNumberField(x^9-3,3^(1/9));
> K meet L;
> Composite(K,L);

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Here is an example to work with towers of fields:

> _<x> := PolynomialRing(Integers());
> suc, K := EmbeddedNumberField(x^2-2,Sqrt(2));
> suc, L := EmbeddedNumberField(Polynomial([-K.1,0,1]),2^(1/4));
> ComplexImage(L.1);
1.18920711500272106671749997056
> ComplexImage(L!(K.1));
1.41421356237309504880168872421

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