Exercise 1, § 5

a) Let E be an extension of a field K, $x \in E$ a transcendental element over K and let $F = K(x^n)$ for an integer $n > 1$. Show that if the intersection $F \cap K[(x-1)(x^n-1)]$ is distinct from K, then K is of characteristic $p > 0$ and $n = p^e$, so that $K(x)$ is a p-radical extension of F. (Writing $y = x-1$ and using Ex. 12, b) of V, p. 149, show that $(y+1)^n - 1 = a y^n$ for an $a \in K$.) b) Show that if $K \subset F' \subset K(x)$ and if $K(x)$ is not a p-radical extension of $F'$ and if $K \neq F' \neq K(x)$, then there exists $z \in K[x]$ such that $z \notin K$ and $F' \cap K[z] = K$. (Argue by contradiction, using a) and Ex. 12, b) of V, p. 149.)

c) Under the same hypotheses as in b), suppose that $F' = K(y)$, where $y \in K[x]$; show that there exists $u \in K[x]$ such that $u \notin K$ and $F' \cap K(u) = F' \cap K[u] = K$. (Observe that if this were false, we should have $F' \cap K[z] \neq K$ for all $z \in K[x]$ such that $z \notin K$, using Ex. 12, c) of V, p. 149.)