Exercise 5.3.2

Proof.

(a)

Let $F$ have characteristic $p$, $p\ne 0$. Then $p$ is prime. Let $\alpha ,\beta \in F$.

If $p$ is an odd prime,

$${(\alpha -\beta )}^p=\alpha +{(-\beta )}^p={\alpha }^p+{(-1)}^p{\beta }^p={\alpha }^p-{\beta }^p.$$

In the remaining case $p=2$, then $1=-1$, thus

$${(\alpha -\beta )}^p={(\alpha +\beta )}^p={\alpha }^p+{\beta }^p={\alpha }^p-{\beta }^p.$$

(b)

Let $H:F\to F,x\mapsto x^p$ the Frobenius homomorphism of $F$. By induction, we show that $H^n(x)=x^{p^n}$ for all $x\in F$ :

$H^0(x)=x=x^{p^0}$, and

$H^n(x)=x^{p^n}\Rightarrow H^{n+1}(x)=H(H^n(x))={(x^{p^n})}^p=x^{p.p^n}=x^{p^{n+1}}$.

If $e\in \mathbb{N}$, as $H^e$, power of a homomorphism, is a homomorphism, so

$$H^e(\alpha +\beta )=H^e(\alpha )+H^e(\beta ),$$

namely

$${(\alpha +\beta )}^{p^e}={\alpha }^{p^e}+{\beta }^{p^e}.$$