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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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We have e'f' = e f , and therefore 0 e l = e , f ' = f , e l ' = f l ' = 1. v$ The ramification index e and the inertia degree f admit a further interesting group-theoretic interpretation. 5) Definition. The kernel IQ C G p of the homomorphism Gp ---+ G(K(Q)IK(P)) is called the inertia group of Q over K . The fixed field T ~ = { x E L ~ ~ xf o=r xa l l a ~ l ~ } is called the inertia field of Q over K. 3 appears in the tower of fields KcZpETp5L7 and we have the exact sequence ) o/p, we of the residue class field O/'J3.

The case where a prime ideal p of K is ramified in L is an exceptional phenomenon. 4) Proposition. 1fL I K is separable, then there are only finitely many prime ideals of K which are ramified in L. This shows that the prime ideals of the ring R = B[X] /(p(X)) are the principal ideals (pi) generated by the pi (X) mod p(X), for i = 1, . . , that the degree [R/(p;) : b] equals the degree of the polynomial pi (X), and r that (01 = (p) = (pilei . nr=, 5 8. , L = K (Q), and let p(X) E o[X] be its minimal polynomial.

Let p I@,,(xnP)for suitable x. Deduce a contradiction from this. 14) and VII, 5 13)). More generally, choosing instead of A \ (0) any nonempty S G A \ (0) which is closed under multiplication, one again obtains a ring structure on the set The most important special case of such a multiplicative subset is the complement S = A \ p of a prime ideal p of A. In this case one writes A , instead of A S - ' , and one calls the ring A p the localization of A at p. When dealing with problems that involve a single prime ideal p of A at a time it is often expedient to replace A by the localization A p .

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Algebraische Zahlentheorie by Jürgen Neukirch


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