# For which fields is the inverse Galois problem known?

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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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? –  Steven Landsburg Mar 3 '12 at 3:34
Yes, thank you. –  Makhalan Duff Mar 3 '12 at 3:36
I fixed it in the body. –  Makhalan Duff Mar 3 '12 at 4:06
The inverse Galois problem is also known to be true for function fields over fields that contain a large field. –  Jérôme Poineau Mar 3 '12 at 8:18
I am no specialist, but I would be very surprised if a field like $k(T)$ could ever be large. –  Jérôme Poineau Mar 4 '12 at 9:16

## 1 Answer

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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