Exercise 8, § 9

Let K be an infinite field, E an algebraic extension of K of finite degree and τ an automorphism of the vector K-space E such that for all x $\in E$, $\tau(x)$ is conjugate to x over K. Show that τ is a K-automorphism of the field E. (Let N be the quasi-Galois extension of K generated by E and let Γ be the group of K-automorphisms of N, A the subgroup of Γ consisting of all E-automorphisms and $\sigma_j$ ($1 \leq j \leq r$) a system of representatives of the left cosets of A. If $(u_i)_{1 \leq i \leq n}$ is a basis of the vector K-space E, consider the polynomial