  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 $\gcd(s,n)=1$.
