A characterization of linearized polynomials with maximum kernel

We provide sufficient and necessary conditions for the coefficients of a q-polynomial f over $F_{q^n}$ which ensure that the number of distinct roots of f in $F_{q^n}$ equals the degree of f. We say that these polynomials have maximum kernel. As an application we study in detail q-polynomials of degree $q^{n−2}$ over $F_{q^n}$ which have maximum kernel and for n ≤ 6 we list all q-polynomials with maximum kernel. We also obtain information on the splitting field of an arbitrary q-polynomial. Analogous results are proved for q^s-polynomials as well, where gcd(s, n) = 1.