All infinite groups are Galois groups over any field

Abstract:

Let $G$ be an arbitrary monoid with $1$ and right cancellation, and $K$ be a given field. We will construct extension fields $F \supseteq K$ with endomorphism monoid End $F$ isomorphic to $G$ modulo Frobenius homomorphisms. If $G$ is a group, then Aut $F = G$. Let ${F^G}$ denote the fixed elements of $F$ under the action of $G$. In the case that $G$ is an infinite group, also ${F^G} = K$ and $G$ is the Galois group of $F$ over $K$. If $G$ is an arbitrary group, and $G = 1$, respectively, this answers an open problem (R. Baer 1967, E. Fried, C. U. Jensen, J. Thompson) and if $G$ is infinite, the result is an infinite analogue of the still unsolved Hilbert-Noether conjecture inverting Galois theory. Observe that our extensions $K \subset F$ are not algebraic. We also suggest to consider the case $K = {\mathbf {C}}$ and $G = \{ 1\}$.