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Books > Science & Mathematics > Mathematics > Algebra
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Divisor Theory (Hardcover, 1990)
Loot Price: R1,623
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Divisor Theory (Hardcover, 1990)
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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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