By Jürgen Neukirch (auth.)

"The current ebook has as its objective to solve a discrepancy within the textbook literature and ... to supply a complete advent to algebraic quantity thought that's principally in accordance with the fashionable, unifying notion of (one-dimensional) mathematics algebraic geometry. ... regardless of this exacting software, the booklet continues to be an creation to algebraic quantity conception for the beginner... the writer discusses the classical innovations from the point of view of Arakelov theory.... The remedy of sophistication box idea is ... really wealthy in illustrating enhances, tricks for extra examine, and urban examples.... The concluding bankruptcy VII on zeta-functions and L-series is one other striking good thing about the current textbook.... The booklet is, with none doubt, the main up to date, systematic, and theoretically complete textbook on algebraic quantity box thought available." W. Kleinert in Z.blatt f. Math., 1992 "The author's enthusiasm for this subject is never as obtrusive for the reader as during this booklet. - an exceptional publication, a gorgeous book." F. Lorenz in Jber. DMV 1995 "The current paintings is written in a truly cautious and masterly style. It doesn't express the trials that it should have triggered even a professional like Neukirch. It definitely is at risk of develop into a vintage; the extra in order fresh advancements were taken under consideration so that it will no longer be outmoded quick. not just needs to it's lacking from the library of no quantity theorist, however it can easily be prompt to each mathematician who desires to get an idea of recent arithmetic." J. Schoissengeier in Montatshefte Mathematik 1994

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**Example text**

E OK, ala and alb = ala ± b; ala = al)a. And an ideal number a should be determined by the totality of its divisors in OK a= {aEOKIaIa}. But in view of the rules for divisibility, this set is an ideal of OK. This is the reason why RICHARD DEDEKIND re-introduced Kummer's "ideal numbers" as being the ideals of OK. Once this is done, the divisibility relation a I a can simply be defined by the inclusion a E a, and more generally the divisibility relation a I b between two ideals by b C a. In what follows, we will study this notion of divisibility more closely.

A, of relatively prime ideals a1, ... , a, , one has an analogue of the well-known "Chinese Remainder Theorem" from elementary number theory. We may formulate this result for an arbitrary ring taking into account that n a = nai. "1 Indeed, since ai I a, i = 1, ... , n, we find on the one hand that a C n =1 ai, and for a E ni ai we find that ai I a, and therefore, the factors being relatively prime, we get a = a1 . , a E a. § 3. 6) Chinese Remainder Theorem. Let at, ... , an be ideals in a ring o such that ai + aj = 0 for i ; j.

Prp, a contradiction. H. Uniqueness of the prime ideal factorization. gs ... be two prime ideal factorizations of a. Then p, divides a factor q i , say q j, and being maximal equals Q1. -. ,r. p'r, vi>0. fl In the sequel such an identity will be automatically understood to signify that the pi are pairwise distinct. If in particular a is a principal ideal (a), then - following the tradition which tends to attribute to the ideals the role of "ideal numbers" - we will write with a slight abuse of notation a = pit ...