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

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Let F be any 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 all exponents are a power of q. We assume that the coefficient of the x term of L is nonzero. We investigate the Galois group G 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.