Exercise 8, § 14

Let E be an extension of a field K. a) Let x be an element of $E - K$. Show that there exists a pure extension $L \subset E$ of K such that E is algebraic over L and $x \notin L$ (consider the set of subextensions L of E which are pure transcendental over K and such that $x \notin L$). b) Let $x \in E$ be a transcendental element over K and p an integer $\geq 2$. Show that there exists a pure transcendental extension $L \subset E$ of K such that E is algebraic over L, $x \notin L$ and $x^p \in L$ (same method). c) Let F be a subextension of E and let $x \in E$ be an algebraic element over F and $P \in F[X]$ its minimal polynomial. Show that there exists a pure extension $L \subset E$ of K such that E is algebraic over L and P is irreducible over the field $F(L)$.