Exercise 1, § 16

Let K be a field of characteristic $p > 0$, E a separable extension of K, of finite transcendence degree over K. a) If there exists a transcendence basis $B_0$ of E over K and an integer $m \geq 0$ such that $K(E^{p^m})$ is separable over $K(B_0)$, show that for every transcendence basis B of E over K there exists an integer $n \geq 0$ such that $K(E^{p^n})$ is separable over $K(B)$. b) Deduce that if the condition in a) holds, E admits a separating transcendence basis over K (if S is a p-basis of E over $K(E^p)$ and B a transcendence basis of E over K containing S (V, p. 176, Ex. 6), show that B is a separating transcendence basis of E over K, using Ex. 6 of V, p. 176). c) Let K be a field of characteristic $p > 0$ and x an element transcendental over K (in an algebraically closed extension $\Omega$ of K). Show that the union E of the extensions $K(x^{p^{-n}})$ of K is a separable extension of K such that $\mathrm{tr.}\deg_K E = 1$ but which does not admit a separating transcendence basis. d) Let K be a perfect field. Show that the perfect closure of the field of rational fractions $K(T)$ is an extension of K of transcendence degree 1 which does not admit a separating transcendence basis.