Exercise 5, § 10

Show that every compact totally disconnected group is isomorphic to the group $\mathrm{Gal}(N/K)$ of a Galois extension of a field (using Gen. Top. III, p. 325, Ex. 3, reduce to the case where the group is a product of finite groups and use Ex. 4 as well as V, p. 59, Example 5).