Exercise 14, § 11

a) Let K be a field such that there exist algebraic extensions of K of arbitrarily large finite degree, and that all these extensions have as degree a power of the same prime number p. Show that under these conditions, for each power $p^h$ ($h \geq 0$) there exists an extension of K of degree $p^h$ (use I, p. 77, Th. 1).

b) Let $p$ be a prime number and $K$ a field such that there exists an algebraic extension of $K$ of degree divisible by $p$ if $p \neq 2$ and divisible by 4 if $p = 2$. Show that there exists an algebraic extension $E$ of $K$ such that the set of degrees of algebraic extensions of $E$ is the set of powers $p^h$ of $p$. (If $K$ is perfect or of characteristic $\neq p$, let $K_1$ be the perfect closure of $K$; consider a maximal algebraic extension of $K_1$, among all those whose elements are of degree not divisible by $p$ over $K$; use a) and Ex. 11.)