Let $L / K$ be a finite extension of fields. Define the trace $\operatorname{Tr}_{L / K}(x)$ and norm $N_{L / K}(x)$ of an element $x \in L$.

Assume now that the extension $L / K$ is Galois, with cyclic Galois group of prime order $p$, generated by $\sigma$.

i) Show that $\operatorname{Tr}_{L / K}(x)=\sum_{n=0}^{p-1} \sigma^{n}(x)$.

ii) Show that $\{\sigma(x)-x \mid x \in L\}$ is a $K$-vector subspace of $L$ of dimension $p-1$. Deduce that if $y \in L$, then $\operatorname{Tr}_{L / K}(y)=0$ if and only if $y=\sigma(x)-x$ for some $x \in L$. [You may assume without proof that $\operatorname{Tr}_{L / K}$ is surjective for any finite separable extension $L / K$.]

iii) Suppose that $L$ has characteristic $p$. Deduce from (i) that every element of $K$ can be written as $\sigma(x)-x$ for some $x \in L$. Show also that if $\sigma(x)=x+1$, then $x^{p}-x$ belongs to $K$. Deduce that $L$ is the splitting field over $K$ of $X^{p}-X-a$ for some $a \in K$.