Exercise 1, § 8

Let K be a field, K' a subfield of K and V a non-zero right vector space over K.

(a) Let V' be a K'-structure on V. Show that K' is equal to the set of $\mu \in K$ such that, for all $x \in V'$, $x\mu \in V'$.

(b) Let $\Gamma'$ be the group of automorphisms of K leaving invariant the elements of K'. Show that if there exists no element of K not belonging to K' and invariant under all the automorphisms $\sigma \in \Gamma'$, there exists no element of V not belonging to V' and invariant under every bijective dimorphism of V (relative to an automorphism of K) which leaves invariant all the elements of V'.

(c) Conversely, let E be a subset of V and G the group of bijective dimorphisms of V leaving invariant all the elements of E; for all $u \in G$, let $\sigma_u$ be the corresponding automorphism of K and let $\Gamma$ be the group of automorphisms of K the image of G under the homomorphism $u \mapsto \sigma_u$. Suppose that G contains no automorphism of V distinct from the identity automorphism and that there exists no element of V not belonging to E and invariant under all the dimorphisms $u \in G$. Show that if K' is the subfield of elements of K invariant under all the automorphism $\sigma \in \Gamma$, E is a K'-structure on V. (Prove first that E is an additive subgroup of V containing a basis of V; let K'' be the set of elements $\mu \in K$ such that the relation $x \in E$ implies $x\mu \in E$. Show that the elements of K'' are invariant under all $\sigma \in \Gamma$ and that K'' is a subfield of K; deduce that E is a K''-structure on V.)