Minimal and characteristic polynomials of symmetric matrices in characteristic two

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