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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?

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    $\begingroup$ The ideas used to make progress on special cases tend to be very interesting in their own right (e.g., see David Zwyina's recent beautiful proof of the Inverse Galois problem for projectivized special orthogonal groups over finite fields of characteristic at least 5), since one needs to find new ways to build extensions with controlled Galois group. That alone is a compelling reason for interest in the topic, even if one doesn't find the problem to be interesting for its own sake. $\endgroup$
    – user74230
    Commented Mar 14, 2015 at 21:45
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    $\begingroup$ possible duplicate of The inverse Galois problem, what is it good for? $\endgroup$
    – abx
    Commented Mar 15, 2015 at 7:52

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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.

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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?

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    $\begingroup$ I think that my question was misunderstood. I do not try to find interest in the question. I am trying to understand the nature of the problem by its consequences. $\endgroup$
    – Ilias Andreou
    Commented Mar 14, 2015 at 23:03
  • $\begingroup$ I understand. I just believe there's no much consequence in the fact that a given group is realizable, or that all of them are (or are not). If there was, it would be well known, given all the attention the problem receives. $\endgroup$
    – Myshkin
    Commented Mar 15, 2015 at 0:02
  • $\begingroup$ By the way, I think the algorithms for calculating Galois groups became much better recently. $\endgroup$
    – Lior Bary-Soroker
    Commented Mar 25, 2015 at 17:33
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You can follow the papers of Professor Debes of lille1 university . He is working on inverse Galois problems http://math.univ-lille1.fr/~pde/

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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."

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