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Let $K$ be a char $0$ field with algebraic closure $\bar K$ and absolute Galois group $G$. Let $\mathcal U$ be an ultrafilter on $\mathbb N$ and $F=\bar K^\mathbb N/\mathcal U$ be the ultrapower of $\bar K$ on $\mathcal U$, which has a natural action of $G$.

What can be said of the field $F^G$ of $G$-invariant points in $F$ ? In particular, can we describe in some way the field extension $F^G/K$ ?

Edit : the afore-mentionned field extension is quite large. It would be more interesting and more natural to look at $F^G/(K^\mathbb N/\mathcal U)$ (the ultrapower of $K$ over $\mathcal U$, which can easily be seen as a subfield of $F^G$).

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up vote 3 down vote accepted

This is quite trivial but not enough for a comment: $F^G$ is not always $K^\mathbb N/\mathcal U$.

Let $x_1,x_2,\dots$ be elements of $K$ such that no product of finitely many of them is a perfect square in $K$. For instance, if $K=\mathbb Q$ we can take them to be the primes.

Our element of $\bar{K}^\mathbb N/\mathcal U$ will come from the following sequence of elements of $\bar{K}$: If $n$ is written in base $2$ as $2^{e_1}+2^{e_2}+\dots + 2^{e_k}$, the $n$th element of the sequence is $\sqrt{x_{e_1}x_{e_2} \dots x_{e_k}}$.

For each finitely generated subgroup of $G$, infinitely many elements of the sequence are invariant under it. This is because any finite codimension subspace of $\mathbb F_2^n$ stil has infinitely many elements, by induction on the codimension.

Thus, the set of all subsets of $\mathbb N$ containing the elements of the sequence invariant under any finitely generated subgroup is a filter, which can be extended to an ultrafilter.

Since for each element of $G$, the subsequence invariant under that element is in the ultrafilter, the associated element of $F$ is $G$-invariant. It lives in an algebraic extension of $G$.

Conversely, if $G$ is topologically finitely generated, then $F^G$ is $K^\mathbb N/\mathcal U$, because being if the set invariant under each generator is in the ultrafilter, the set invariant under all the generators is in the ultrafilter, and that's also the set of elements of $K$.

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Thanks a lot! This is actually helpful for my original problem where $K$ was a $p$-adic field, and otherwise very interestingly gives an example with non-trivial extension. Do you have any idea of what this extension may or may not be in general (algebraic, finite, ...) ? –  Cyrille Corpet Jun 27 '13 at 22:50
My guess is that it doesn't have those nice properties. One can easily modify this construction to use degree $n$ abelian extensions, and so construct an element in $F^G% of degree $n$ over $K^{\mathbb N}/\mathcal U$. It's not immediately obvious to me that you can do these all with the same ultrafilter but it's hard to see how you'd be unable to, so the degree is infinite. Then you can pull the same trick with a very slowly increasing degree to get transcendentals. –  Will Sawin Jun 28 '13 at 17:12
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