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Let $K$ be a field and $G$ be a group of its automorphisms endowed with thecompact-open topology. There are many situations, where it is natural to studythe category $Sm_K(G)$ of smooth (i.e. with open stabilizers) $K$-semilinearrepresentations of $G$. The category $Sm_K(G)$ is semisimple (in which case $K$is a generator of $Sm_K(G)$) if and only if $G$ is precompact. In this note westudy the case of the non-precompact group $G$ of all permutations of aninfinite set $S$. It is shown that the categories $Sm_K(G)$ are locallynoetherian; the morphisms are `locally split'. Given a field $F$ and a subfield$k\neq F$ algebraically closed in $F$, one of principal results describes theGabriel spectra (and related objects) of the categories $Sm_K(G)$ for some of$G$-invariant subfields $K$ of the fraction field $F_{k,S}$ of the tensorproduct over $k$ of the labeled by $S$ copies of $F$. In particular, the object$F_{k,S}$ turns out to be an injective cogenerator of the category $Sm_K(G)$for any $G$-invariant subfield $K$ of $F_{k,S}$. As an application, whentranscendence degree of $F|k$ is 1, a correspondence between the $G$-invariantsubfields of $F_{k,S}$ algebraically closed in $F_{k,S}$ and certain systems ofisogenies of `generically $F$-pointed' torsors over absolutely irreducibleone-dimensional algebraic $k$-groups is constructed, so far only incharacteristic 0. The only irreducible finite-dimensional smooth representationof $G$ is trivial. However, there is an invariant `cross-ratio' subfield $K$ of$k(S)$ such that the irreducible finite-dimensional smooth $K$-semilinearrepresentations of $G$ correspond to the irreducible algebraic representationsof $PGL_{2,k}$. In general, any smooth $G$-field $K$ admits a smooth $G$-fieldextension $L|K$ such that $L$ is a cogenerator of $Sm_L(G)$.

Invariant fields of rational functions and semilinear representations of symmetric groups over them