### Article

## 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 de-

scription: (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].