The Inverse Galois Problem and minimal ramification over function fields.

摘要

For many years, efforts have been made to solve the Inverse Galois Problem, concerned with finding an extension of a given field K having a given Galois group, with special emphasis placed on the case where K = Q . Here we consider the particular case where the base field is K = Fp (t) and conjecture that, for a given finite group G, there is a G-extension and that, moreover, we can place minimality conditions on the extension. Specifically, we consider the numbered of ramified primes in the field extension, and how it relates to properties of the group itself. We make a conjecture on the existence of G-extensions of Fp (t) and give a conjectural formula for the minimal number of primes, both finite and infinite, ramified in G-extensions, and give theoretical and computational proofs for many cases of this conjecture.