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I was reading through chapter 14 of Dummit and Foote just now and I came across the sentence "It is an open problem to determine whether every finite group appears as the Galois group for some polynomial over \mathbb{Q}" and I was wondering 2 things:

  1. Is it known whether each finite group is the Galois group of some polynomial (not necessarily over $\mathbb{Q}$)?

  2. Why is this problem so hard? (I know this question may defy a simple answer, but from the very naive perspective of someone who knows very little--me--it isn't very clear why this should be too much harder than the abelian case considering what is known about the structure of finite groups in general)

thanks :)

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    $\begingroup$ For (1), see math.stackexchange.com/questions/132665/…. See also en.wikipedia.org/wiki/Inverse_Galois_problem. $\endgroup$
    – lhf
    May 10 '16 at 1:56
  • $\begingroup$ My naive understanding of your second question is that 1) Abelian groups have a much simpler structure. That is, given an abelian number field, EVERY subextension is Galois because every subgroup of the Galois group is normal! This is wildly false for non-abelian number fields. 2) We do not understand Artin $L$-functions at all as well as we do Dirichlet $L$-functions. The proof I've seen of the Inverse Galois problem for abelian groups is using Dirichlet's theorem on primes in Arithmetic progressions which is proved using the latter. $\endgroup$
    – RKD
    May 10 '16 at 2:08
  • $\begingroup$ @lhf Thanks! You found that reference fast! $\endgroup$
    – user140776
    May 10 '16 at 2:24
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Yes, every finite group is the Galois group of some extension. Consider the field $K = \mathbb{Q}(e_1, \dots e_n)$ and its extension $L = \mathbb{Q}(x_1, \dots x_n)$ where the $x_i$ satisfy

$$\prod (t - x_i) = \sum (-1)^i e_i t^{n-i}$$

where $e_0 = 1$, so the $e_i$ are the elementary symmetric polynomials in the $x_i$. Then $L$ is a finite Galois extension of $K$ with Galois group $S_n$ (exercise). Since every finite group $G$ embeds in some $S_n$, we get finite Galois extensions $L$ of fields $L^G$ with Galois group $G$.

The inverse Galois problem is hard not because we don't know enough about finite groups but because we don't know enough about the Galois theory of $\mathbb{Q}$. It's equivalent to showing that every finite group is a quotient of the absolute Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, and this is just hard. The abelian case is much easier because abelian quotients of the Galois group are handled abstractly by class field theory and concretely by the Kronecker-Weber theorem, but "nonabelian class field theory" is much harder, not completely understood, and very much the subject of active research.

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