Let \(p\) be a prime, \(F_p\) the field with \(p\) elements and \(K=F_p(t)\) the field of rational functions in a variable \(t\). An additive polynomial \(f\in K[x]\) is a polynomial of the form \(f(x)=\sum_{i=0}^na_ix^{p^i}\). The zeros of \(f\) form a vector space over \(F_p\) and hence for any \(g|f\), the Galois group of \(g\) is contained in \(\text{GL}_n(F_p)\). This observation is used to bound the Galois group of a polynomial from above. Knowledge on the action of the Frobenius automorphism and on transitivity is used to obtain a lower bound.
As a result the authors find polynomials with Galois groups \(M_{24}, M_{23}\), \(\text{PSL}_3(3)\), \(\text{PSL}_2(7)\), \(\text{PSL}_3(2)\) and \(D_4\) over \(F_2(t)\) and \(M_{11}\) over \(F_3(t)\). For some of those polynomials, the roots can be analyzed using Taylor series or Puiseux series. This is done for the polynomials \(\text{Fano}(x)=x^7+tx+1\in F_2(t)[x]\), \(\text{Mathieu}(x)=x^{24}+x+t\in F_2(t)[x]\) and \(s_{11}(x)=x^{11}+tx^2-1 \in F_3(t)[x]\) that have Galois groups \(\text{PSL}_3(2)\), \(M_{24}\) and \(M_{11}\) over \(F_3(t)\), respectively. The polynomial \(\text{Fano}(x)\) is shown to be closely related to the polynomial \(x^7+tx^3+1\) that by a theorem of Serre has Galois group \(\text{PSL}_3(2)\).