Definition 2.0.1. Jargon watch.
By a place of a number field we mean either an archimedean completion or (an infinite place), or a -adic completion for some nonzero prime ideal of (a finite place).
Each place corresponds to an equivalence class of absolute values on if is a place, we write for the corresponding completion, which is either or for some prime (This means that two complex embeddings of which differ by complex conjugation are considered the same place of )
This form of parity between finite and infinite places will be a recurring theme throughout this book.