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,  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 .