By Jürgen Neukirch
Algebraische Zahlentheorie: eine der traditionsreichsten und aktuellsten Grunddisziplinen der Mathematik. Das vorliegende Buch schildert ausführlich Grundlagen und Höhepunkte. Konkret, sleek und in vielen Teilen neu. Neu: Theorie der Ordnungen. Plus: die geometrische Neubegründung der Theorie der algebraischen Zahlkörper durch die "Riemann-Roch-Theorie" vom "Arakelovschen Standpunkt", die bis hin zum "Grothendieck-Riemann-Roch-Theorem" führt.
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Additional resources for Algebraische Zahlentheorie (Springer-Lehrbuch Masterclass)
If n e:: m, then we view f as defined on Xn through the natural projection on X m • We conclude at once from the compatibility relation that 2: f(x)CPn(x) xeX Il = 2: f(x)CPm(x). xeXm Let X be the projective limit with the limit topology, so that X is a compact space. For each n we have a surjective map For each x E Xn the inverse image r,;-l(x) is an open set in X, and the totality of such open sets for all n, x is a basis for the topology of X. A functionf on X is called locally constant if and only if there exists il such that f factors through X n • Such functions are also called step functions, and their group is denoted by St(X, K).
In the next sections we shall deal with Bernoulli numbers systematically. For the moment, we need only a special case, so we define ad hoc the first Bernoulli polynomial and the first Bernoulli number BI = on Z(m) we define f BI,f = --1-, its constant term. 3 can be reformulated as follows. 1. xlx annihilates ~(P)CX). 15 1. Character Sums Corollary 1. Assume that m = p is prime ::::: 3. If X is not equal to the Teichmuller character and is non-trivial, then ord BI'ilx = ord Bl,i,' Proof Immediate from the lemma and the theorem.
4 if you don't find this obvious. Lemma 5. (8- R : 8- Rm() = ± m M f1 xodd B1 •Z • Proof First observe that the sign is whatever is needed to make the righthand side positive. Multiplication by ,;-m() is an endomorphism of QR-, which is a semisimple algebra, decomposing into a product of I-dimensional algebras corresponding to the odd characters. X' X odd On the other hand, ,; - m() maps ,; - R into itself, and by standard elementary linear algebra, the index is given by the absolute value of the determinant.
Algebraische Zahlentheorie (Springer-Lehrbuch Masterclass) by Jürgen Neukirch