Minimal ramification and the inverse Galois problem over the rational function field Fp(t)Fp(t)

• We explore the inverse Galois problem over the rational function field.
• We give a conjecture and evidence for the minimal number of ramified primes.
• We provide a proof of many cases as well as computational examples.

The inverse Galois problem is concerned with finding a Galois extension of a field K   with given Galois group. In this paper we consider the particular case where the base field is K=Fp(t)K=Fp(t). We give a conjectural formula for the minimal number of primes, both finite and infinite, ramified in G-extensions of K, and give theoretical and computational proofs for many cases of this conjecture.