Divisor Theory by Harold M. Edwards

By Harold M. Edwards

Guy 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. examine at the present time such a lot mathematicians who find out about Kronecker's thought of divisors learn about it from having learn Hermann Weyl's lectures on algebraic quantity idea [We], and regard it, as Weyl did, in its place to Dedekind's concept of beliefs. Weyl's axiomatization of what he calls "Kronecker's" idea is built-as Dedekind's thought was once built-around distinct issue ization. besides the fact that, in proposing the idea during this approach, Weyl overlooks one in all Kronecker's most dear rules, specifically, the concept that the target of the idea is to outline maximum com mon divisors, to not in achieving factorization into primes. the explanation Kronecker gave maximum universal divisors the first position is straightforward: they're self reliant of the ambient box whereas factorization into primes isn't really. The very proposal of primality depends upon the sphere below consideration-a leading in a single box could consider a bigger field-so if the idea is based on factorization into primes, extension of the sector includes a very new thought. maximum universal divisors, however, could be outlined in a way that doesn't swap in any respect whilst the sphere is prolonged (see 1.16). purely after he has laid the basis of the speculation of divisors does Kronecker think of factorization of divisors into divisors best in a few targeted box.

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A process will be given which produces either an [h] with the required properties or a proper refinement of the factorization of [g], that is, a refinement in which at least one [gil is given as a product of at least two integral divisors i= [1]. 17 shows that it must eventually produce an [h]. ', where the [gd are relatively prime integral divisors =I [1] and the O"i are positive integers. Let [g] = [gl][g2]· .. [gIL]. For each i = 1,2, ... ,p, let hi be a polynomial with coefficients in J{ such that [hi] is the product of [f] and all the [gj] other than [gil; in other words, let [hd = [f][g][gd- 1 .

In the same way, tP n + ¢>n is relatively prime to B. Therefore, a is finite at AB for all sufficiently large n . ¢>n-l("Y_(3)/(tP n +¢>n), which is ¢> times an element of K finite at A for large n. Therefore, a - (3 is zero at A for large n, that is, a and (3 represent the same elements of VA. In the same way, a and "Y represent the same elements of VB for all sufficiently large n, so a has the required property when n is large. 24 Rings of Values 43 (4) If a represents a nonzero element of VA, then the denominator of [a] is relatively prime to A and its numerator is not divisible by A.

PROPOSITION (9). [ilia for all coefficients a of f. This is the final proposition in the basic theory of divisors. The other propositions have all been deduced from elementary properties of natural rings and algebraic extensions of them. This last one depends, however, on a less elementary theorem. It can be deduced, for example, from Corollary 4 of the theorem of Part 0: PROOF: If f = 0, there is nothing to prove. Otherwise, by (8), fq = C1T for some nonzero c E r and for some primitive polynomial 1T with coefficients in r.

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