Exercise 3, § 10

Let N be a Galois extension of a field K of infinite degree over K. Show that the cardinal of the Galois group $\mathrm{Gal}(N/K)$ is greater than that of the set of subsets of N (use V, p. 61 and 62). Deduce that there exist non-closed subgroups of $\mathrm{Gal}(N/K)$.