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Working paper

An analogue of Hilbert's Theorem 90 for infinite symmetric groups

Let K be a field and G be a group of its automorphisms. If K is algebraic over the subfield KG fixed by G then, according to Hilbert's Theorem 90, any smooth (i.e. with open stabilizers) K-semilinear representation of the group G is isomorphic to a direct sum of copies of K. If K is not algebraic over KG then there exist non-semisimple smooth semilinear representations of G over K, so Hilbert's Theorem 90 does not hold. Let now G be the group of all permutations of an infinite set S acting naturally on the field k(S) freely generated over a subfield k by the set S. The goal of this note is to present three examples of G-invariant subfields K\subseteq k(S) such that the smooth K-semilinear representations of G of {\sl finite length} admit an explicit description, close to Hilbert's Theorem 90. Namely, (i) if K=k(S) then any smooth K-semilinear representation of G of finite length is isomorphic to a direct sum of copies of K, (ii) if K\subset k(S) is the subfield of rational homogeneous functions of degree 0 then any smooth K-semilinear representation of G of finite length splits into a direct sum of one-dimensional K-semilinear representations of G, (iii) if K\subset k(S) is the subfield generated over k by x-y for all x,y\in S then there is a unique isomorphism class of indecomposable smooth K-semilinear representations of G of each given finite length.