# Finite Fields

June 8, 2022

### Abstract

This entry formalizes the classification of the finite fields (also called Galois fields): For each prime power $p^n$ there exists exactly one (up to isomorphisms) finite field of that size and there are no other finite fields. The derivation includes a formalization of the characteristic of rings, the Frobenius endomorphism, formal differentiation for polynomials in HOL-Algebra and Gauss' formula for the number of monic irreducible polynomials over finite fields: $\frac{1}{n} \sum_{d | n} \mu(d) p^{n/d} \textrm{.}$ The proofs are based on the books from Ireland and Rosen, as well as, Lidl and Niederreiter.