Jürgen Neukirch's Algebraische Zahlentheorie PDF

February 27, 2018 | Number Theory | By admin | 0 Comments

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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Let S denote a finite set of prime ideals of (not any more a multiplicative subset), and let X be the set of all prime ideals that do not belong to S. We put The units of this ring are called the S-units, and the group C I = ~~l(oi) the S-class group of K . 7) Corollary. For the group K~ = (o;)* of S-units of K there is an isomorphism K S 2 p ( ~ ) Z#S+r+s-l where I . and s are defined as in Q 5 , p. 30. Proof: The torsion subgroup of K~ is the group p ( K ) of roots of unity in K . 4): rank(^ ') = rank(ok) is induced by mapping + rank( @ Z)= #S + r + s - 1.

E K*. Because of unique prime factorization, this means that up(@) = 0 for p E X, and vp(ctp) = vp(a) for p $ X. It follows that a! E 0; = o(X)* and a! = ap mod o;. This shows exactness in the middle. 8) Corollary. The S-class group ~ 1 : = Cl(o;) is finite. Exercise 1. Let A be an arbitrary ring, not necessarily an integral domain, Ict M be an A-module and S a multiplicatively closed subset of A such that 0 6 S. In M x S consider the equivalence relation ( m , s) -- (m', s') 3 s" E S such that sl'(s'm - - sm') =0.

1), which we had proved only in the case of a separable extension L J K , is valid for general finite extensions of the field of fractions of a Dedekind domain. Chapter I. Algebraic Integers 78 Next we want to compare the one-dimensional noetherian integral domain o with its normalization 8. 8) provided we make the following hypothesis: ! I 5 12. Orders 79 Observing that P(R) 2 K*/R* for any integral domain R with field of fractions K , we obtain the commutative exact diagram (*) o is an integral domain whose normalization 6 is a finitely generated o-module.

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Algebraische Zahlentheorie by Jürgen Neukirch

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