We provide sufficient and necessary conditions for the coefficients of a $q$-polynomial $f$ over $\mathbb{ F}_{ q^n}$ which ensure that the number of distinct roots of $f$ in $\mathbb{ F}_{ q^n}$ 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 $\mathbb{ F}_{ q^n}$ which have maximum kernel and for $n\leq 6$ 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 $\gcd(s,n)=1$.
Categories: math.CO math.RA