Minimal and characteristic polynomials of symmetric matrices in\n characteristic two

Let $k$ be a field of characteristic two. We prove that a non constant monic\npolynomial $f\\in k[X]$ of degree $n$ is the minimal/characteristic polynomial\nof a symmetric matrix with entries in $k$ if and only if it is not the product\nof pairwise distinct inseparable irreducible polynomials. In this case, we\nprove that $f$ is the minimal polynomial of a symmetric matrix of size $n$. We\nalso prove that any element $\\alpha\\in k_{alg}$ of degree $n\\geq 1$ is the\neigenvalue of a symmetrix matrix of size $n$ or $n+1$, the first case happening\nif and only if the minimal polynomial of $\\alpha$ is not the product of\npairwise distinct inseparable irreducible polynomials.\n