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By Jürgen Neukirch

Die algebraische Zahlentheorie ist eine der traditionsreichsten und gleichzeitig heute besonders aktuellen Grunddisziplinen der Mathematik. In dem vorliegenden Buch wird sie in einem ausf?hrlichen und weitgefa?ten Rahmen abgehandelt, der sowohl die Grundlagen als auch ihre H?hepunkte enth?lt. Die Darstellung f?hrt den Leser in konkreter Weise in das Gebiet ein, l??t sich dabei von modernen Erkenntnissen ?bergeordneter Natur leiten und ist in vielen Teilen neu. Der grundlegende erste Teil ist mit einigen neuen Aspekten versehen, wie etwa einer ausf?hrlichen Theorie der Ordnungen. ?ber die Grundlagen hinaus enth?lt das Buch eine geometrische Neubegr?ndung der Theorie der algebraischen Zahlk?rper durch die Entwicklung einer "Riemann-Roch-Theorie" vom "Arakelovschen Standpunkt", die bis zu einem "Grothendieck-Riemann-Roch-Theorem" f?hrt, ferner lokale und globale Klassenk?rpertheorie und schlie?lich eine Darstellung der Theorie der Theta- und L-Reihen, die die klassischen Arbeiten von Hecke in eine fa?liche shape setzt.

Das Buch wendet sich an Studenten nach dem Vordiplom bzw. Bachelor. Dar?ber hinaus ist es dem Forscher als weiterweisendes Handbuch unentbehrlich.

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And Grunwald did so. The reference “Grunwald [2]” in Hasse’s [Has:1933] refers to Grunwald’s paper, at that time still “forthcoming”, which appeared 1933 in Crelle’s Journal [Gru:1933]. There Grunwald proved a general existence theorem which became known as “Grunwald’s theorem”. This theorem is much stronger than Hasse’s Existence Theorem: Grunwald’s theorem Let K be an algebraic number field and S a finite set of primes of K. For each p ∈ S let there be given a cyclic field extension Lp|Kp . Moreover, let n ∈ N be a common multiple of the degrees [Lp : Kp] .

This is easy enough. For, let S denote the set of those primes p of K for which the local index mp of D is = 1. Choose π ∈ K which is a prime element for every finite p ∈ S, and π < 0 for every √ p infinite √ real p ∈ S. Then K( π ) splits D by the Splitting Theorem, hence K( p π ) is isomorphic to a subfield of D. Applying Albert’s theorem it follows that D is cyclic. Hasse continues: We are trying to generalize your theorem to prime power degree. This would eliminate Grunwald’s theorem altogether for the proof of the Main Theorem.

We have said above that this proof is remarkable. This does not mean that the proof is difficult; in fact, it is straightforward for anyone who is acquainted with Hensel’s method of handling valuations. The remarkable thing is that Hasse used valuations to investigate non-commutative division algebras over local fields. 41 The valuation ring of Dp consists of all x ∈ Dp with v(x) ≥ 0. It contains a unique maximal ideal, which is a 2-sided ideal, consisting of all x with v(x) > 0. 41 After Hasse, the valuation theory of non-commutative structures developed rapidly, not only over number fields but over arbitrary fields.

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