HILBERT'S THEOREM 90 FOR NON-COMPACT GROUPS

Let K be a field and G be a group of its automorphisms. It follows from Speiser's generalization of Hilbert's Theorem 90, [10] that any K-semilinear representation of the group G is isomorphic to a direct sum of copies of K, if G is finite. In this note three examples of pairs (K, G) are presented such that certain irreducible K-semilinear representations of G admit a simple description: (i) with precompact G, (ii) K is a field of rational functions and G permutes the variables, (iii) K is a universal domain over field of characteristic zero and G its automorphism group. The example (iii) is new and it generalizes the principal result of [7].