CUBO A Mathematical Journal Vol.12, N°02, (97–121). June 2010

The tree of primes in a field

Wolfgang Rump

Institute for Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany email: rump@mathematik.uni-stuttgart.de

Dedicated to B. V. M.

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.

Key words and phrases: prime, valuation, product formula.

RESUMEN

La fórmula producto de la Teoría de Números Algebraicos conecta primos finitos e infinitos de una formula estricta, un hecho no difícil de ser verificado, es que nunca cesa de ser estudiado. Nosotros proponemos un nuevo concepto de primos para cualquier cuerpo e investigamos algunas de sus propiedades. Hay primos algebraicos, correspondientes a valuaciones, talque todo primo contiene un primo algebraico mayor. Para un número de cuerpos, esta parte algebraica es cero solamente para primos infinitos. Es demostrado que los primos de cualquier cuerpo forman un árbol con una clase de estructura auto-similar, hay una operación binaria sobre los primos inexplorada incluso para los racionales. Todo primo define una topología sobre el cuerpo, y todo primo compacto da origen a una única medida de Haar, jugando rol esencial en la fórmula producto.

AMS (MOS) Subj. Class.: Primary: 11S15, 11N80, 12J20, 13A18. Secondary: 28C10

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Received: November 2008.

Revised: March 2009.