# Standard Lattices of Compatibly Embedded Finite Fields

## TL;DR

This paper introduces a new lattice construction for compatibly embedded finite fields that is more efficient than existing methods like Bosma-Cannon-Steel, with practical implementation demonstrating its effectiveness.

## Contribution

It generalizes existing constructions of finite field lattices using the Lenstra-Allombert isomorphism algorithm, offering improved efficiency and practicality.

## Key findings

- Larger set of field extensions from small tables
- Quadratic complexity algorithms
- Linear storage relative to number of extensions

## Abstract

Lattices of compatibly embedded finite fields are useful in computer algebra systems for managing many extensions of a finite field $\mathbb{F}_p$ at once. They can also be used to represent the algebraic closure $\bar{\mathbb{F}}_p$, and to represent all finite fields in a standard manner. The most well known constructions are Conway polynomials, and the Bosma-Cannon-Steel framework used in Magma. In this work, leveraging the theory of the Lenstra-Allombert isomorphism algorithm, we generalize both at the same time. Compared to Conway polynomials, our construction defines a much larger set of field extensions from a small pre-computed table; however it is provably as inefficient as Conway polynomials if one wants to represent all field extensions, and thus yields no asymptotic improvement for representing $\bar{\mathbb{F}}_p$. Compared to Bosma-Cannon-Steel lattices, it is considerably more efficient both in computation time and storage: all algorithms have at worst quadratic complexity, and storage is linear in the number of represented field extensions and their degrees. Our implementation written in C/Flint/Julia/Nemo shows that our construction in indeed practical.