Download e-book for kindle: Algebraische Zahlentheorie by Jürgen Neukirch

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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1) Proposition. The mappings are mutually inverse 1-1 -correspondences between the prime ideals q 5 A \ S of A and the prime ideals 8 of AS-'. Chapter I. Algebraic Integers 66 Proof: If q 5 A I \ 1%1 q E q, s E S) 0 0' is a prime ideal of A S - ' . d = implies that s"aal = qss' E q . Therefore aa' E q , because ss' S" ' s" $! Furthermore one has q=OnA, since = a E 12 f l A implies q = a s E q , whence a E q because s $! q. Conversely, let 12 be an arbitrary prime ideal of AS-'. Then q = 12 fl A is obviously a prime ideal of A , and one has q g A \ S.

As the characteristic p of o / p does not divide n, the polynomials Xn - 1 and nxn-' have no common root in alp. So Xn - 1 mod p has no multiple roots. We therefore see that passing to the quotient o -+ o / p maps the group pn of n-th roots of unity bijectively onto the group of n-th roots of unity of alp. In particular, the primitive n-th root of unity ( modulo p remains a primitive n-th root of unity. The smallest 'extension field of IFp = Z / p Z containing it is the field F p f p ,because its multiplicative group P* is cyclic of order p f p - l .

It is of the most beautiful simplicity. 3) Proposition. Let n = pVp be the prime factorization of n and, for every prime number p, Jet fp be the smallest positive integer such that pfp E 1 mod n/pVp Then one has in Q (0the factorization Multiplying this by h and substituting the result ho = h20 obtain x 2 0 Z[(] = 0. + + h Z [ < ] ,we p = ( p , . . pr)"(pYP), where p , , . . , pr are distinct prime ideals, all of degree ,f,,. Iterating this procedure, we find h t o + Z [ ( ] = o for all t 2 1 .

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