Exercise 5, § 16

a) Let K be a field of characteristic $p > 0$ and E an extension of K admitting a separating transcendence basis over K. Show that the intersection L of the fields $K(E^{p^n})$, where $n$ runs over N, is an algebraic extension of K. (Consider first the case where $\operatorname{tr. deg}_K E < +\infty$; show that L is relatively perfect over K, hence E is separable over L. Now use Ex. 2 and Ex. 6 of V, p. 176. In the general case, if B is a separating transcendence basis of E over K, for each $x \in L$, there is a finite subset B, of B such that the minimal polynomial of $x$ over each of $K(B_0^{p^n})$ is the same as its minimal polynomial over $K(B)$; now deduce that if $F = K(B_0)(x)$, then $x \in K(F^{p^n})$ for all n.) b) Show that the largest perfect field $E^{p^\infty}$ (intersection of the $E^{p^n}$) contained in E is an algebraic extension of $K^{p^\infty}$ (use a) and Ex. 8 of V, p. 176). Give an example where E is a finitely generated extension of a perfect field K but $E^{p^\infty}$ is not contained in K.