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.

Contents

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 FldNum_fldemb-eg-1 (H36E12)

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

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);

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
V2.29, 21 October 2025