Exercise 4, § 10

Let N be a Galois extension of a field K, $(E_i)_{i \in I}$ a family of subextensions of N, Galois over K, such that : 1" for each index i, if $F_i$ is the field generated by the $E_i$, for $j \neq i$, then $E_i \cap F_i = K$; 2" N is generated by the union of the $E_i$. a) Show that N is isomorphic to the tensor product $\bigotimes_{i \in I} E_i$ (III, p. 470) (define an isomorphism of this tensor product onto N, using Th. 5 of V, p. 71). b) Show that $\mathrm{Gal}(N/K)$ is isomorphic to the product of the topological groups $\mathrm{Gal}(E_i/K)$.