Exercise 7.21

Proof. Let $F:\{\begin{array}{ccc}\hfill {𝔽}_q\hfill & \hfill \to \hfill & \hfill {𝔽}_q\hfill \\ \hfill x\hfill & \hfill \mapsto \hfill & \hfill x^p\hfill \end{array}.$

As the characteristic of ${𝔽}_q$ is $p$, ${(x+y)}^p=x^p+y^p$ et ${(\mathit{xy})}^p=x^p{y}^p$, and each homomorphism of field is injective, so $F$ is a field automorphism (Frobenius automorphism).

For every automorphism $H$ in ${𝔽}_q$, and every polynomial $p(x)=\sum <!--\; FUNCTION\; APPLICATION\; -->a_i{x}^i\in {𝔽}_q[x]$, write $(H\cdot p)(x)={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i}H(a_i)x^i$. Then for all $(p,q)\in {𝔽}_q{[x]}^2$, $H\cdot (\mathit{pq})=(H\cdot p)(H\cdot q)$.

With this notation,

Since $\alpha \in {𝔽}_{p^n}$, $F^n\alpha ={\alpha }^{p^n}=\alpha$ , thus

In other words, if $f(x)={\sum <!--\; FUNCTION\; APPLICATION\; -->}_i{a}_i{x}^i$, then for all $i$, $F(a_i)=a_i$, so $a_i^p=a_i$, thus $a_i\in {𝔽}_p$, and $f\in {𝔽}_p[x]$. In particular, the coefficients $a_{n-1}=\alpha +{\alpha }^p+\cdots +{\alpha }^{p^{n-1}},a_0=\alpha {\alpha }^p{\alpha }^{p^2}\cdots {\alpha }^{p^{n-1}}$ are in ${𝔽}_p$. □