We next give the statements of the principal results of class field theory, with almost no proofs. Our goal at this point is to clarify what the statements say and how they can be applied. We will have to discuss plenty of other material before returning to the proofs, but the reader who wishes to peek ahead for a glimpse of the strategy is directed to Section 5.4.
Definition2.0.1.Jargon watch.
By a place of a number field \(K\text{,}\) we mean either an archimedean completion \(K \hookrightarrow \RR\) or \(K \hookrightarrow \CC\) (an infinite place), or a \(\gothp\)-adic completion \(K \hookrightarrow K_\gothp\) for some nonzero prime ideal \(\gothp\) of \(\gotho_K\) (a finite place). (Note: there is only one place for each pair of complex embeddings of \(K\text{.}\))
Each place corresponds to an equivalence class of absolute values on \(K\text{;}\) if \(v\) is a place, we write \(K_v\) for the corresponding completion, which is either \(\RR\text{,}\)\(\CC\text{,}\) or \(K_\gothp\) for some prime \(\gothp\text{.}\)
This form of parity between finite and infinite places will be a recurring theme throughtout this book.
