Exercise 7, § 12

Let K be a finite field, E an algebraic extension of degree $n$ of K and $(a_1, \ldots, a_n)$ a normal basis of E over K ; suppose that the cyclic Galois group $\mathrm{Gal}(E/K)$ is generated by the cyclic permutation $\sigma = (\alpha_1\alpha_2 \ldots \alpha_n)$. Let $\tau$ be an automorphism of the vector K-space E such that for each $x \in E$, $\tau(x)$ is conjugate to x over E (cf. V, p. 154, Ex. 8). a) Show that if K has at least 3 elements, $\tau$ is a K-automorphism of the field E (reduce to the case where $\tau(\alpha_1) = \alpha_1$ and consider $\tau(a, + c\alpha_i)$ for $c \in K$).

b) If $K = F_2$, show that for each element $\rho \in \mathrm{Gal}(E/K)$, the relations $\rho(\alpha_r) = \alpha_r$, and $\rho(\alpha_s) = \alpha_s'$ imply $r' - r \equiv s' - s \pmod{n}$. Deduce that if $n > 5$, $\tau$ is a K-automorphism of the field E. (Note that if $\tau(\alpha_1) = a$, and $\tau(\alpha_1 + \alpha_h) = a_+ + \alpha_k$ with $k \neq h$, we must have $h + k = n + 2$; deduce from this that necessarily $\tau(\alpha_2) = \alpha_n$ and $\tau(\alpha_4) = \alpha_4$ or $\tau(\alpha_4) = \alpha_{n-2}$ which leads to a contradiction.) c) For $K = F_2$ and $3 \leq n \leq 5$ the following automorphisms of the vector K-space E are not field automorphisms of E:

for $n = 3, \tau : (\alpha_1, \alpha_2, \alpha_3) \mapsto (\alpha_1, \alpha_3, \alpha_2)$ for $n = 4, \tau : (\alpha_1, \alpha_2, \alpha_3, \alpha_4) \mapsto (\alpha_1, \alpha_4, \alpha_3, \alpha_2)$ for $n = 5, \tau : (\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5) \mapsto (\alpha_1, \alpha_5, \alpha_4, \alpha_3, \alpha_2)$

but are such that $\tau(x)$ is conjugate to x for all $x \in E$.