TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 304. Number 1. November 1987 ALL INFINITE GROUPS ARE GALOIS GROUPS OVER ANYFIELD MANFRED DUGAS AND RÜDIGER GÖBEL Dedicuted to Bertrum Huppert, on the occasion of his 6Qth birthday on October 22. 1987 Abstract. Let G be an arbitrary monoid with 1 and right cancellation, and A" be a given field. We will construct extension fields F D K with endomorphism monoid End F isomorphic to G modulo Frobenius homomorphisms. If G is a group, then Aut F = G. Let FG denote the fixed elements of F under the action of G. In the case that G is an infinite group, also FG = 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 c F are not algebraic. We also suggest to consider the case K = C and G = {1}. 1. Introduction. We want to investigate field extensions R of an arbitrarily fixed field K of characteristic char K = p a prime or p = 0. Using some terminology from model theory, we will show that field extensions of K are a nonstructure theory in a very strong sense. The extension field R can be chosen quite arbitrary up to the obvious restrictions. Let us recall the related well-known facts. End R will denote the set of all endomorphisms of R. The product of endomorphisms makes End R = (EndR)' to a monoid, which is a set G' with associative multiplication "■" and identity 1 g G. A monoid G satisfies the law of right cancellations if yx = zx for x, y, z g G implies y — z. The endomorphisms of R are necessarily injections, hence End R satisfies the law of right cancellations. Observe that maps are acting from the right. Apparently not all endomorphisms are surjective. If p # 0, the field R always has a special semigroup 0 of such endomorphisms, the Frobenius homomorphisms. The cyclic group <ï> ç End R is generated by ex(p): R -» R which takes r g R to rp. Frobenius homomorphisms are surjective if and only if R is a perfect field. If p = 0, then O0 = (1} = 1. It is the object of this paper to prove the converse of these well-known facts. Theorem. Let K be any field. Then the following conditions are equivalent. (1) G' is a monoid with right cancellation. Received by the editors October 14, 1986. Presented at the Annual AMS Meeting, San Antonio, January 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 20F29, 12A55, 12F20; Secondary 20C99, 20B27, 20F28, 20E36. 355 ©1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use