A characterization of linearized polynomials with maximum kernel

We provide sufficient and necessary conditions for the coefficients of a q-polynomial f over Fqn which ensure that the number of distinct roots of f in Fqn equals the degree of f. We say that these polynomials have maximum kernel. As an application we study in detail q-polynomials of degree qn−2 over Fqn 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 qs-polynomials as well, where gcd⁡(s,n)=1.