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Lattices of compatibly embedded finite fields are useful in computer algebrasystems 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 knownconstructions are Conway polynomials, and the Bosma-Cannon-Steel framework usedin Magma. In this work, leveraging the theory of the Lenstra-Allombertisomorphism algorithm, we generalize both at the same time. Compared to Conwaypolynomials, our construction defines a much larger set of field extensionsfrom a small pre-computed table; however it is provably as inefficient asConway polynomials if one wants to represent all field extensions, and thusyields no asymptotic improvement for representing $\bar{\mathbb{F}}_p$.Compared to Bosma-Cannon-Steel lattices, it is considerably more efficient bothin computation time and storage: all algorithms have at worst quadraticcomplexity, and storage is linear in the number of represented field extensionsand their degrees. Our implementation written in C/Flint/Julia/Nemo shows thatour construction in indeed practical.

Standard Lattices of Compatibly Embedded Finite Fields