Exercise 13, § 14

Let D be a field, V a left vector space over D, A the ring End$_D$(V), and $f$ an automorphism of A. Prove that there exist an automorphism $\sigma$ of D and a bijective mapping $u: V\rightarrow V$ that is semilinear with respect to $\sigma$ (II, §1, No. 13, p. 223) such that we have $f(a) =u\circ a\circ u^{-1}$ for every $a\in A$. (Let $V^f$ be the A-module with additive group V and external law defined by $(a, v)\mapsto f(a)(v)$; deduce from Proposition 4 of VIII, p. 50 an A-isomorphism $u: V\rightarrow V^f$. Then, observe that $uDu^{-1}$ is equal to the bicommutant of the D-module V, that is, to D.)