Class field theory is about the abelian extensions of a number field ${K}$. Actually, this is strictly speaking global class field theory (there is an analog for abelian extensions of local fields), and there is a similar theory for function fields of transcendence degree 1 over finite fields, but we shall not deal with it.

Let us, however, consider the situation for local fields—which we will later investigate more—as follows. Suppose ${k}$ is a local field and ${L}$ an unramified extension. Then the Galois group ${L/k}$ is isomorphic to the Galois group of the residue field extension, i.e. is cyclic of order ${f}$ and generated by the Frobenius. But I claim that the group ${k^*/NL^*}$ is the same. Indeed, ${NU_L = U_K}$ by a basic theorem about local fields that we will prove using abstract nonsense later (but can also be easily proved using successive approximation and facts about finite fields). So ${k^*/NL^*}$ is cyclic, generated by a uniformizer of ${k}$, which has order ${f}$ in this group. Thus we get an isomorphism

$\displaystyle k^*/NL^* \simeq G(L/k)$

sending a uniformizer to the Frobenius. (more…)