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…)