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Several years ago I attended a colloquium talk of an expert in Galois theory. He motivated some of his work on its relation with the inverse Galois problem. During the talk, a guy from the audience asked: ``why should I, as a number theorist, should care about the inverse Galois problem?"
I must say that as a young graduate student that works on Galois theory, I was amazed or even perhaps shocked from this question. But later, I realized that I should have asked myself this question long ago.

Can you pose reasons to convince a mathematician (not just number theorist) of the importance of the inverse Galois problem? or maybe why it's unimportant if you want to ruin the party ;)

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I'm not sure, by the way, that this should be community wiki. –  Pete L. Clark Nov 29 '09 at 8:31
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Almost 200 years after Galois, and we still don't know which groups arise as the Galois group of an irreducible polynomial over Q; and someone asks why should we care??? –  JS Milne Jan 5 '10 at 5:17
    
I am a layman in this subject, but I am curious to know even if people were able to show every simple group is realisable over Q, how far are we from solving the actual inverse Galois problem? –  Ying Zhang May 2 '10 at 19:38
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some simple group are known to be realizable over Q and there are some very recent discoveries too (e.g. Gabor's usage of modular representations), but we are very far from realizing all simple groups. for example the Matthieu group M11 is not known to be realizable (if I am not mistaken). –  Lior Bary-Soroker Jun 1 '10 at 21:19
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You're thinking of $M_{23}$. It's known by now that all the other sporadic simple groups arise, and for many of them, including $M_{11}$, there are explicit examples or even explicit families of examples. –  Noam D. Elkies Jan 12 '13 at 5:36

8 Answers 8

For me, it's one of those questions that would not be so interesting if the answer is Yes but which would probably be very interesting if the answer is No. If not all groups are Galois groups over Q, then there is probably some structure that can be regarded as an obstruction, and then this structure would probably be essential to know about. For instance, not all groups are Galois groups over local fields -- they have to be solvable. This is by basic properties of the higher ramification filtration, which is, surprise, essential to know about if you want to understand local fields. So you could say it's an approach to finding deeper structure in the absolute Galois group. Why not just do that directly? The problem with directly looking for structure is that it's not a yes/no question, and so sometimes you lose track of what exactly you're doing (although in new and fertile subjects often you don't). So the inverse Galois problem has the advantage of being a yes/no question and the advantage that things would be really interesting if the answer is No. Unfortunately, I think the answer is expected to be Yes, though correct me if I'm wrong.

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The question is not really my business, but I can give a stock answer. It might be a good answer in the sense that it convinces an outsider like me.

The narrowest version of the inverse Galois problem, find all of the Galois groups of finite extensions of $\mathbb{Q}$, might not be all that interesting. A better question would be the following: Let $G$ be a finite group and let $\mathbb{F}$ be a field of characteristic 0 (or more generally a perfect field). Can you describe the set (or moduli space if you like) of all Galois extensions of $\mathbb{F}$ over $G$? For instance if $G = C_2$, it's a good question with a good answer; the question is a model of taking square roots of elements of $\mathbb{F}$. With that special case in mind, it's always a good question. It can be viewed as a theory of nonabelian surds.

If for a given field $\mathbb{F}$ and a given finite group $G$, you don't even know if there are any points in the moduli space of extensions with Galois group $G$, then you hardly know anything. In particular, $\mathbb{Q}$ is an important field, and there are many specific finite groups for which people don't even know that much.

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The previous answers are all on point; let me just say a little more.

First, the IGP as a problem is a sink, not a source (or something with both inward and outward flow!): I know of no nontrivial consequences of assuming that every finite group over Q (or even over every Hilbertian field) is a Galois group. This does not mean it's a bad problem: the same holds for Fermat's Last Theorem.

As with FLT, if IGP were easy to prove, then it would be of little interest. (As a good example, if you know Dirichlet's theorem on primes in arithmetic progressions, it's easy to prove that every finite abelian group occurs as a Galois group of Q. What does this tell you about the maximal abelian extension of Q? Not much -- the Kronecker-Weber theorem is an order of magnitude deeper.) But as with FLT, the special cases of IGP that have been established use a wide array of fascinating techniques and provide an important border-crossing between algebra and geometry.

Arguably more interesting than IGP itself is the Regular Inverse Galois Problem: for any field K and any finite group G, there exists a regular function field K(C)/K(t) with Galois group isomorphic to G. (If K is Hilbertian -- e.g. a global field -- then RIGP for K implies IGP for K.) Now RIGP is of great interest in arithmetic geometry: given any finite group G there are infinitely many moduli spaces (Hurwitz spaces) attached to the problem of realizing G regularly over K (because we have discrete invariants which can take infinitely many possible values, like the number of branch points). If even one of these Hurwitz schemes has a K-rational point, then G occurs regularly over K. In general, the prevailing wisdom about varieties over fields like Q is that they should have very few rational points other than the ones that stare you in the face. (Yes, it is difficult or impossible to formalize this precisely.) So it is somewhat reasonable to say that the chance that a given Hurwitz space -- say of general type -- has a Q-rational point is zero, but what about the chance that at least one of infinitely many Hurwitz spaces, related to each other by various functorialities, has a Q-rational point? To me that is one of mathematics' most fascinating questions: to learn the answer either way would be tremendously exciting.

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"So it is somewhat reasonable to say that the chance that a given Hurwitz space -- say of general type, as most of them are -- has a Q-rational point is zero" Just a note: what "most" means here may depend strongly on how you count. When G is fixed and your discrete invariants (like number of branch points) vary, it's not at all clear that most of the resulting Hurwitz spaces are general type! –  JSE Nov 29 '09 at 15:18
    
Corrected accordingly. Thanks. –  Pete L. Clark Nov 29 '09 at 21:24

I personally know of no immediate applications of a positive (or negative) answer to the inverse Galois problem. At the same time, the problem seems to me a useful standard against which to gauge mathematical progress.

Answering the inverse Galois problem for solvable extensions required class field theory (one of the pinnacles of early 20th century mathematics). This can be seen as evidence that the ability to solve the inverse Galois problem will entail a deeper understanding of a variety of mathematical things.

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I agree with you. It is similar to Fermat last problem. The equation is just one of a gazillion others. But still it drove crazy the mathematical society for centuries, and was a catalysis for many very interesting mathematics. You can give the inverse Galois problem the credit for Hilbert's irreducibility theorem, which is one of my favorites. –  Lior Bary-Soroker Nov 25 '09 at 2:18
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I also agree. In fact, the problem is more natural than other famous open problems so it is an obvious challenge mathematicians face and progress in mathematics measured. There are views (see ihes.fr/~gromov/topics/SpacesandQuestions.pdf ) that dismiss the importance of "natural" problems rather than deep emerging problems. But even if you agree to this opinion (and I tend not to agree) Natural old-standing problems stand as an objective measure for progress in math. –  Gil Kalai Nov 28 '09 at 19:59
    
when you say “Answering the inverse Galois problem for solvable extensions required class field theory”, what do you mean? do you mean solvable extension over Q or over local fields? Thank you. –  natura Jan 5 '10 at 4:51
    
I also agree that IGP is a very natural problem. Galois theory is one of the very first (highly) nontrivial example of equivalence of seemingly different categories, and it's one the most beautiful. –  natura Jan 5 '10 at 4:54

Bauer's Theorem (a simple consequence of the Chebotarev Density Theorem) states that a finite Galois extension K of an algebraic number field F is uniquely determined (as a subield of some fixed algebraic closure of F) by the set of primes of F which split completely in K. Thus knowing all possible Galois groups is the same as knowing all possible splitting laws in finite Galois extensions. Being able to describe these splitting laws in some explicit fashion is basically "nonabelian reciprocity", which is THE most important problem in algebraic number theory, so the "inverse Galois problem" is of FUNDAMENTAL importance to all number theorists.

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Any branch of mathematics after the first few definitions will make everyone routinely ask themselves some basic questions. I consider Inverse Galois Problem is one such. If the question (i) is not highly technical, (ii) can be understood at very early stages and (iii) does not sound concocted then it justifies itself.

These are the natural questions the subject should attempt to answer. (It is irrelevant if solving them requires Fields medallists or undergraduates).

Let me list more questions in the same category (not necessarily of the same level of difficulty!)

  1. Which divisors of $|G|$ are orders of subgroups of $G$?

  2. Which connected open subsets of the complex plane are biholomorphic to the unit disc?

  3. For which numbers $d$, is the ring $\mathbf{Z}[\sqrt d]$ a UFD?

  4. Which finite groups occur as subgroups of $\mathbf{SO}(3)$?

  5. Which integers are represented by an indefinite/definite integral quadratic form?

  6. Which projective curves are subvarieties of the projective plane?

I have been under the impression that this is the way mathematicians think. If someone questions the relevance of the above questions it would be difficult for me to communicate with that person.

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I agree that the inverse Galois problem is interesting by itself. However, it seems to me that it is a sink, as Pete formulated it. Many of the questions you raise above, have many beautiful implication to other topics in math, and outside of math. Thus making them interesting in a much deeper sense. –  Lior Bary-Soroker Jan 13 '13 at 6:35

just a link, Families of number fields of prime discriminant if we not only consider about the group, but also put some other restrictions on the extension (such as discriminant, ramifications etc), then we have the above problem, which is quite interesting.

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P.S. Infinitely many Galois extensions inside an algebraic closure can have isomorphic Galois groups, so the "explicit" description of the primes which split in a finite Galois extension is the crux of of the matter when it comes to nonabelian reciprocity.

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