Generic Polynomials and Frobenius Modules

The Inverse Galois Problem is concerned with determining whether there exists, for a given finite group G and field K, a Galois extension L/K, such that the Galois group Gal(L/K) is isomorphic to G. An approach to solving this problem is through the use of generic polynomials. Specifically, we say a polynomial f in K(t₁,…, t_m)[X] is G-generic if its Galois group over K(t₁,…, t_m) is G and every Galois G-extension L/M, where M contains K, is the splitting field of some specialization f(a₁,…,a_m, X), where a₁,…, a_m are in M. As generic polynomials provide a complete solution of the Inverse Galois Problem, their construction is worthwhile.

In this presentation, we consider the group of units G = E^* of any finite dimensional associative algebra E over the finite field F_q, and demonstrate that we can always construct explicit G-generic polynomials f in F_q(t₁,…, t_m)[X], where m is less than or equal to dim_Fq(E), using Matzat's theory of Frobenius Modules.