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The inverse Galois problem is known for (or in Jarden's and Fried's terminology, the following fields are universally admissible) function fields over henselian fields (like $\mathbb{Q}_p(x)$); function fields over large fields (like $\mathbb{C}(x)$); and large Hilbertian fields (conjecturally $\mathbb{Q}^{ab}$, although I'm not certain that any field is known to be in this category).

Clarification:

A large field $K$ (a.k.a. an ample field) is a field such that if $V$ is a variety of dimension $\geq 1$ over $K$ with at least one smooth $K$-rational point, then it has infinitely many smooth $K$-rational points. For example any algebraically closed field is large.

A Hilbertian field is more difficult to explain, but it suffices to say that any number field and any function field (over any field) is Hilbertian.

My question is:

Is there a proof (not a conjecture) that there exists a field $K$ which is neither a function field over a henselian field, nor a function field over a large field, nor a large Hilbertian field, such that the inverse Galois problem is true over that field? (i.e. that every finite group is realizable as a Galois group over that field)

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  • $\begingroup$ You wrote: "A large field K (a.k.a. an ample field) is a field such that if V is a variety over K with at least one K -rational point, then it has infinitely many K -rational points. For example any algebraically closed field is large." I trust this is missing some condition on V. Did you mean to assume that it's positive-dimensional? $\endgroup$ Mar 3 '12 at 3:34
  • $\begingroup$ I fixed it in the body. $\endgroup$ Mar 3 '12 at 4:06
  • $\begingroup$ The inverse Galois problem is also known to be true for function fields over fields that contain a large field. $\endgroup$ Mar 3 '12 at 8:18
  • $\begingroup$ Also, I am a bit surprised by the examples you give at the beginning since henselian fields are large. $\endgroup$ Mar 3 '12 at 8:26
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    $\begingroup$ I am no specialist, but I would be very surprised if a field like $k(T)$ could ever be large. $\endgroup$ Mar 4 '12 at 9:16
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You should find what you want in the following article by Jochen Koenigsmann: The regular inverse Galois problem over non-large fields. J. Europ. Math. Soc., 6(4):425–434, 2004.

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