The tree of primes in a field

Abstract

The product formula of algebraic number theory connects finite and infinite primes in a stringent way, a fact, while not hard to be checked, that has never ceased to be tantalizing. We propose a new concept of prime for any field and investigate some of its properties. There are algebraic primes, corresponding to valuations, such that every prime contains a largest algebraic one. For a number field, this algebraic part is zero just for the infinite primes. It is shown that the primes of any field form a tree with a kind of self-similar structure, and there is a binary operation on the primes, unexplored even for the rationals. Every prime defines a topology on the field, and each compact prime gives rise to a unique Haar measure, playing an essential part in the product formula.