Exercise 1, § 17

Let K_0 be a field of characteristic p > 0, K = K_0(X, Y) the field of rational fractions in two indeterminates over K_0, U and V two indeterminates, $\Omega$ an algebraically closed extension of K(U, V), E = K(U, u), where $u^p = X + YU^p$ and F = K(V, o), where $v^p = X + YV^p$; E and F are not separable over K (V, p. 178, Ex. 6, c)). a) Show that K is relatively algebraically closed in E and in F (if x ∈ E is algebraic over K, show that we must have $x^p \in K$; if $x \notin K$, then E = K(U, x), deduce that then $X^{1/p}$ and $Y^{1/p}$ would lie in K(x)). b) Show that E and F are linearly disjoint over K but that K is not relatively algebraically closed in K(E U F). (Prove that $X^{1/p} \in K(E U F)$; deduce that v cannot belong to E(V), and conclude from this that E and F are linearly disjoint over K.)