# Finite field embeddings

## Introduction

Nemo allows the construction of finite field embeddings making use of the algorithm of Bosma, Cannon and Steel behind the scenes to ensure compatibility. Critical routines (e.g. polynomial factorization, matrix computations) are provided by the C library Flint, whereas high level tasks are written directly in Nemo.

## Embedding functionality

It is possible to explicitly call the embedding `embed`

function to create an embedding, but it is also possible to directly ask for the conversion of a finite field element `x`

in some other finite field `k`

via calling `k(x)`

. The resulting embedding is of type `FinFieldMorphism`

. It is also possible to compute the preimage map of an embedding via the `preimage_map`

function, applied to an embedding or directly to the finite fields (this actually first computes the embedding), or via conversion. An error is thrown if the element you want to compute the preimage of is not in the image of the embedding.

### Computing an embedding

`Nemo.embed`

— Method`embed(k::T, K::T) where T <: FinField`

Embed $k$ in $K$, with some additional computations in order to satisfy compatibility conditions with previous and future embeddings.

**Examples**

```
julia> k2, x2 = finite_field(19, 2, "x2")
(Finite field of degree 2 and characteristic 19, x2)
julia> k4, x4 = finite_field(19, 4, "x4")
(Finite field of degree 4 and characteristic 19, x4)
julia> f = embed(k2, k4)
Morphism of finite fields
from finite field of degree 2 and characteristic 19
to finite field of degree 4 and characteristic 19
julia> y = f(x2)
6*x4^3 + 5*x4^2 + 9*x4 + 17
julia> z = k4(x2)
6*x4^3 + 5*x4^2 + 9*x4 + 17
```

### Computing the preimage of an embedding

`AbstractAlgebra.Generic.preimage_map`

— Method`preimage_map(k::T, k::T) where T <: FinField`

Computes the preimage map corresponding to the embedding of $k$ into $K$.

`AbstractAlgebra.Generic.preimage_map`

— Method`preimage_map(f::FinFieldMorphism)`

Compute the preimage map corresponding to the embedding $f$.

`preimage_map(f::FinFieldPreimage)`

Compute the preimage map corresponding to the preimage of the embedding $f$, i.e. return the embedding $f$.

**Examples**

```
julia> k7, x7 = finite_field(13, 7, "x7")
(Finite field of degree 7 and characteristic 13, x7)
julia> k21, x21 = finite_field(13, 21, "x21")
(Finite field of degree 21 and characteristic 13, x21)
julia> s = preimage_map(k7, k21)
Preimage of a morphism
from finite field of degree 7 and characteristic 13
to finite field of degree 21 and characteristic 13
julia> y = k21(x7);
julia> z = s(y)
x7
julia> t = k7(y)
x7
```