Exercise 9, § 16

With the notation of Ex. 8 let M ⊂ L be a subextension. Put m = tr. deg_kM, n = tr. deg_kL, r = n - m. Let x_1, ..., x_r in L be such that L is a separable algebraic extension of k(x_1, ..., x_r) = K. a) Show that, after renumbering the x_i if necessary, we may suppose that L is algebraic over M(x_1, ..., x_r). b) Show that there exists a finitely generated subextension N ⊂ M such that: a) L is algebraic over N(x_1, ..., x_r); β) K' = K(N(x_1, ..., x_r)) is a finite extension of N(x_1, ..., x_r); γ) the canonical homomorphism K' $\otimes_{N(x_1, ..., x_r)} M(x_1, ..., x_r) \to K(M(x_1, ..., x_r)) = K''$ is bijective. c) Let N be a subextension as in b). Show successively that K'' is a separable algebraic extension of K' (because L is separable over K), that M(x_1, ..., x_r) is a separable algebraic extension of N(x_1, ..., x_r) (use b), γ)) and that M is a separable algebraic extension of N. d) Show that M is a separable algebraic extension of a pure transcendental extension (use c) and the fact that N is a finitely generated separable extension of k).