Matrix Representations of Finite Fields

Abstract

Finite fields are important algebraic structures that have a wide range of applications in fields such as coding theory and cryptography. But the standard construction of finite field extensions through polynomial quotients is computationally opaque, especially when we want to identify a degree-$2$ extension of $F_8$ and a degree-$3$ extension of $F_4$. In this short note, we present a coherent family of representations by matrices $ρ_q^n\colon F_{q^n} \to F_q^{n\times n}$ for all prime powers $q$ and all degrees $n \ge 1$. These maps are chosen so that concatenating $ρ_{q^n}^m$ and $ρ_q^n$ recovers $ρ_q^{nm}$ up to row and column permutations. As a consequence, the images of $ρ_2^6$ can be partitioned into four $3 \times 3$ blocks or nine $2 \times 2$ blocks to visualize the subfield chains $F_{64} / F_8 / F_2$ and $F_{64} / F_4 / F_2$ at the same time. A variant $\varrho$ is also discussed, wherein the Frobenius automorphism is represented by a cyclic shift of rows and columns. From an educational point of view, these rhos give explicit and self-contained mental models of finite fields; subfields, trace, norm, minimal polynomial, and Frobenius all become visible through matrix algebra accessible to most students. From a theoretical point of view, the construction exhibits structural implications of Conway polynomials and the normal basis theorem.