Minimal and characteristic polynomials of symmetric matrices in characteristic two

Abstract

Let k be a field of characteristic two. We prove that a monic polynomial f∈k[X] of degree n≥1 is the minimal/characteristic polynomial of a symmetric matrix with entries in k if and only if it is not the product of pairwise distinct inseparable irreducible polynomials. In this case, we prove that f is the minimal polynomial of a symmetric matrix of size n. We also prove that any element α∈kalg of degree n≥1 is the eigenvalue of a symmetric matrix of size n or n+1, the first case happening if and only if the minimal polynomial of α is separable.