Consequences of the Inverse Galois Problem Are there any papers written about the consequences of the Inverse Galois Problem in case it is proved to be true or false?
We know a lot of things that would be true if the Riemann Hypothesis holds. What results would the Inverse Galois problem imply?
 A: I once saw an application of a solved case of the inverse Galois problem.
It is well known, that the Dedekind $\zeta$-function of a number field does not determine the number field up to isomorphy. In the talk it was shown that the $\zeta$-function together with a certain number of twists by characters do determine the number field. Let $K$ be the number field in question, $L$ be its normal closure. To define the right twist an abelian extension $M$ of $K$ was considered, which is as independent from $L$ as possible, that is, the Galois group of the normal closure of $M$ is a wreath product of the Galois group of $L$ and a cyclic group. The existence of such an $M$ is a special case of the inverse Galois problem, which had been solved before.
Sorry, but I have no name or further detail.
A: I don't think there's any big consequence of the Inverse Galois problem being answered either way. If the beautiful mathematics behind it and the influential methods (rigidity) are not a good enough reason, here is a very down-to-earth explanation of why should we care about its solution, taken from "Groups as Galois Groups", by Helmut Volklein:

The idea of encoding algebraic-arithmetic information in terms of
  group theory was the beginning of both Galois theory and group theory.
  [...] One of the aspects of the theory that remains unsatisfactory is
  the fact that it is very hard to compute the Galois group of a given
  polynomial. Therefore, the full correspondence between equations of
  degree $n$ and subgroups of $S_n$ can only be worked out for very
  small values of $n$. Since it is probably impossible to get a full
  understanding of this correspondence for general $n$, one is naturally
  led to the following more reasonable question: Do at least all
  subgroups of $S_n$ occur in this correspondence?

A: You can follow the papers of Professor Debes of lille1 university . He is working on inverse Galois problems http://math.univ-lille1.fr/~pde/
A: See The inverse Galois problem, what is it good for?
Clark says "I know of no nontrivial consequences of assuming that every finite group over Q is a Galois group."
