Exercise 3, § 12

Let K be a finite field of q elements ; for every prime number l let N, be the union of the extensions of K (contained in an algebraic closure $\Omega$ of K) whose degree is a power of l. Show that N, is an abelian extension of K whose Galois group is isomorphic to the group $\mathbf{Z}_l$ of l-adic integers. Deduce that $\Omega$ is isomorphic to the tensor product $\bigotimes_l N_l$, and hence, that $\mathrm{Gal}(\Omega/K)$ is isomorphic to the product $\prod_l \mathbf{Z}_l$.