Algebraische Zahlentheorie by Jürgen Neukirch

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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 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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This shows that O is noetherian, provided L 1 K is separable. We ask the reader's permission to content ourselves for the time being with this case. We shall come back to the general case on a more convenient occasion. 8)). + < Exercise 4. Let be a primitive p-th root of unity, p an odd prime number. Show c k ( l +)" 1 0 5 k < 5, n E Z], that Z[(]* = (<)Z[< +<-'I*. Show that Z[+]* = (f if p = 5. + < Exercise 5. Let be a primitive rn-th root of unity, rn >_ 3. Show that the numbers for (k. The subgroup of the group of units they generate is called the group of cyclotomic units.

I ( 0 ) G @ P ( o p ) . 211 P Identifying the subgroup P (o)with its image in the direct sum one gets because it is the biggest ideal such that aa-' g o. 4) Proposition. A fractional ideal a of o is invertible if and only if, for every prime ideal p # 0, ap = a o p is a fractional principal ideal of op. 4), and we have ap = upfor almost all p because a lies in only finitely many maximal ideals p. We therefore obtain a homomorphism Chapter I. , ap E pop # up. In order to prove surjectivity, let (apop) E P ( o p ) be given.

Putting s = $1 . . s,, , we have sai E a p o for i = 1, . , n, hence sap'a 5 o and therefore sap' E a - ' . Consequently, s = sa;'ap E a-'a E p, a contradiction. 0 We denote the group of invertible ideals of o by J ( o ) . It contains the group P ( o ) of fractional principal ideals a o , a E K*. 5) Definition. The quotient group 0 is called the Picard group of the ring o. For the ring o , the fractional ideals of o , in other words, the finitely generated nonzero o-submodules of the field of fractions K , no longer form a group - unless o happens to be Dedekind.

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