Minimal and characteristic polynomials of symmetric matrices in characteristic two

Let k be a field of characteristic two. We prove that a monic polynomial f is an element of 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 alpha is an element of k(alg) 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 alpha is separable. (c) 2021 Elsevier Inc. All rights reserved.