Exercise 6, § 16

Let K be a field of characteristic $p > 0$, E an extension of K such that there exists an integer $h \geq 0$ for which $K(E^{p^h})$ is separable over K (which is always the case when E is a finitely generated extension of K). a) If C is a p-basis of E over $K(E^p)$, there is a subset B of C such that $B^{p^h}$ is a p-basis of $K(E^{p^h})$ over $K(E^{p^{h+1}})$; B is an algebraically free set over K. Show that the extension $F = K(E^{p^h})(B)$ is separable over K and E is an algebraic extension of F. b) Show that B is ap-basis of F over $K(F^p)$. Show that if $x \in E$ and if $m$ is the least integer such that $x^{p^m} \in F$, then $x^{p^m} \in K(F^{p^m})$; therefore E is a subfield of the field $K^{p^{-\infty}}(F)$ (isomorphic to $K^{p^{-\infty}} \otimes_K K$). A subextension F of E is said to be distinguished if it is separable over K and if E is a subfield of $K^{p^{-m}}(F)$; F is then a maximal separable extension of K in E. c) Let P be a perfect field of characteristic $p > 0$, $K = P(X, Y)$ the field of rational fractions in two indeterminates and $\Omega$ an algebraically closed extension of the field of rational fractions $K(Z)$ over $K$. If $u \in \Omega$ is such that $u^p = X + Y Z^p$, show that $E = K(Z, u)$ is an extension of $K$ such that $K(Z)$ and $K(u)$ are two distinguished subextensions of $E$, so that there exists no larger separable subextension of $E$. If $v \in \Omega$ is such that $v^{p^2} = X + Y Z^p$, show that in the extension $E' = K(Z, v)$, $K(Z)$ is a maximal separable subextension of $E$ but is not a distinguished subextension.