EAN: 9780817634483

DIVISOR THEORY - HAROLD M EDWARDS GEBUNDEN

Man sollte weniger danach streben  die Grenzen der mathe­ matischen Wissenschaften zu erweitern
als vielmehr danach  den bereits vorhandenen Stoff aus umfassenderen Gesichts­ punkten zu
betrachten - E. Study Today most mathematicians who know about Kronecker's theory of divisors
know about it from having read Hermann Weyl's lectures on algebraic number theory [We]  and
regard it  as Weyl did  as an alternative to Dedekind's theory of ideals. Weyl's axiomatization
of what he calls Kronecker's theory is built-as Dedekind's theory was built-around unique
factor­ ization. However  in presenting the theory in this way  Weyl overlooks one of
Kronecker's most valuable ideas  namely  the idea that the objective of the theory is to define
greatest com­ mon divisors  not to achieve factorization into primes. The reason Kronecker gave
greatest common divisors the primary role is simple: they are independent of the ambient field
while factorization into primes is not. The very notion of primality depends on the field under
consideration-a prime in one field may factor in a larger field-so if the theory is founded on
factorization into primes  extension of the field entails a completely new theory. Greatest
common divisors  on the other hand  can be defined in a manner that does not change at all when
the field is extended (see §1.16). Only after he has laid the foundation of the theory of
divisors does Kronecker consider factorization of divisors into divisors prime in some
specified field.

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