Download Algebraische Zahlentheorie by Jürgen Neukirch PDF

By Jürgen Neukirch

ISBN-10: 0124859674

ISBN-13: 9780124859678

Die algebraische Zahlentheorie ist eine der traditionsreichsten und gleichzeitig heute besonders aktuellen Grunddisziplinen der Mathematik. In dem vorliegenden Buch wird sie in einem ausführlichen und weitgefaßten Rahmen abgehandelt, der sowohl die Grundlagen als auch ihre Höhepunkte enthält. Die Darstellung führt den Studenten in konkreter Weise in das Gebiet ein, läßt sich dabei von modernen Erkenntnissen übergeordneter Natur leiten und ist in vielen Teilen neu. Der grundlegende erste Teil ist mit einigen neuen Aspekten versehen, wie etwa der "Minkowski-Theorie" und einer ausführlichen Theorie der Ordnungen. Über die Grundlagen hinaus enthält das Buch eine geometrische Neubegründung der Theorie der algebraischen Zahlkörper durch die Entwicklung einer "Riemann-Roch-Theorie" vom "Arakelovschen Standpunkt", die bis zu einem "Grothendieck-Riemann-Roch-Theorem" führt, ferner eine moderne Darstellung der Klasssenkörpertheorie und schließlich eine neue Theorie der Theta-Reihen und L-Reihen, die die klassischen Arbeiten von Hecke in eine faßliche shape setzt. Das Buch ist an Studenten nach dem Vorexamen gerichtet, darüber hinaus wird es sehr bald dem Forscher als weiterweisendes Handbuch unentbehrlich sein.

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E 8 r7 0 = a, so that 1 a = a . , E as-'. , oS-' is noetherian. + SI of the field of fractions. The significance of these valuations lies in their relation to the prime ideal factorization. If x E K * and is the prime factorization of the principal ideal ( x ) ,then, for each p, one has up = vp ( x ) . Sn with coefficients $ E OS-', then multiplying it with the n-th power of s = sl . . sn shows that sx is integral over 0, whence s x E o and therefore x E oS-I. This shows that oS-' is a Dedekind domain.

Let A be an integral domain. If the localization As is integral over A . then As = A . Exercise 7 (Nakayama's Lemma). Let A be a local ring with maximal ideal m, let M be an A-module and N c M a submodule such that M I N is finitely generated. Then one has the implication: 3 12. Orders is an order. The theoretical significance of orders, however, lies in the fact that they admit "singularities", which are excluded as long as only Dedekind domains with their "regular" localizations opare considered.

5) Proposition. Let f! and p be odd prime numbers, C* = (- 1) a primitive f! -th root of unity. Then one has: < p is totally split in Q ( & ) C, and p splits in Q ( { ) into an even number of prime ideals. Proof: The little computation in $ 8 , p. 51 has shown us that C* = t 2with t = CaE(z/lz,* SO that ~ ( f ii )Q ( { ) . If p is totally split in (f)<", ~ ( f isay ) ,p = ~ 1 ~ then 2 , some automorphism a of Q ( { ) such that a p l = p2 transforms the set of all prime ideals lying above p l bijectively into the set of prime ideals above p2.

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