  Let $K$ be a field and $G$ be a group of its automorphisms. If $G$ is
precompact then $K$ is a generator of the category of smooth (i.e. with open
stabilizers) $K$-semilinear representations of $G$. There are non-semisimple
smooth semilinear representations of $G$ over $K$ if $G$ is not precompact.
  In this note the smooth semilinear representations of the group $G$ of all
permutations of an infinite set $S$ are studied. Let $k$ be a field and $k(S)$
be the field freely generated over $k$ by the set $S$ (endowed with the natural
$G$-action). One of principal results describes the Gabriel spectrum of the
category of smooth $k(S)$-semilinear representations of $G$.
  It is also shown, in particular, that (i) for any smooth $G$-field $K$ any
smooth finitely generated $K$-semilinear representation of $G$ is noetherian,
(ii) for any $G$-invariant subfield $K$ in the field $k(S)$, the object $k(S)$
is an injective cogenerator of the category of smooth $K$-semilinear
representations of $G$, (iii) if $K\subset k(S)$ is the subfield of rational
homogeneous functions of degree 0 then there is a one-dimensional
$K$-semilinear representation of $G$, whose integral tensor powers form a
system of injective cogenerators of the category of smooth $K$-semilinear
representations of $G$, (iv) 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.
