Class field theory is about the abelian extensions of a number field . 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 is a local field and an unramified extension. Then the Galois group is isomorphic to the Galois group of the residue field extension, i.e. is cyclic of order and generated by the Frobenius. But I claim that the group is the same. Indeed, 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 is cyclic, generated by a uniformizer of , which has order in this group. Thus we get an isomorphism
sending a uniformizer to the Frobenius. (more…)