On Galois groups of linearized polynomials related to the special linear group of prime degree

Let $F$ be a field of prime characteristic $p$ and let $q$ be a power of $p$. We assume that $F$ contains the finite field of order $q$. A $q$-polynomial $L$ over $F$ is an element of the polynomial ring $F[x]$ with the property that those powers of $x$ that occur as terms of $L$ with nonzero coefficient have exponent a power of $q$. If the exponent of the leading term of $L$ is $q^n$, we say that $L$ has $q$-degree $n$. We assume that the coefficient of the $x$ term of $L$ is nonzero. We investigate the Galois group $G$, say, of $L$ over $F$, under the assumption that $L(x)/x$ is irreducible in $F[x]$. It is well known that if $L$ has $q$-degree $n$, its roots are an $n$-dimensional vector space over the field of order $q$ and $G$ acts linearly on this space. Our main theorem is the following. We consider a monic $q$-polynomial $L$ whose $q$-degree is a prime, $r$, say, with $L(x)/x$ irreducible over $F$. Assuming that the coefficient of the $x$ term of $L$ is $(-1)^r$, we show that the Galois group of $L$ over $F$ must be the special linear group $SL(r,q)$ of degree $r$ over the field of order $q$ when $q>2$. We also prove a projective version for the projective special linear group $PSL(r,q)$, and we present the analysis when $q=2$.