Exercise 3, § 5

Let $K$ be a field of characteristic $p > 2$ and let $F$ be the field of rational fractions $K(X, Y)$. Let $E = F(\theta)$ be an extension of $F$ generated by a root $\theta$ of the polynomial

of $F[Z]$. Show that $[E : F(\theta^p)] = p$, but that $E$ contains non-radical element over $F$ not in $F$. (Firstly remark that $f$ is irreducible in $F[Z]$; if $\beta \in E$ existed such that $\beta^p \in F$, $\beta \notin F$, $f$ would be reducible in $F(\beta)[Z]$; using Ex. 2, show that then $X^{1/p}$ and $Y^{1/p}$ would belong to $E$ and we would have $[E : F] \geq p^2$.) (See Ex. 1, V, § 7).