Von Fermat bis Minkowski: Eine Vorlesung über Zahlentheorie und ihre Entwicklung (German Edition) Review

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Von Fermat bis Minkowski: Eine Vorlesung über Zahlentheorie und ihre Entwicklung (German Edition) ReviewThe book begins with two cursory chapters, "The Beginnings" and Fermat. Chapter 3 introduces Euler's analytic number theory (zeta function, determination of zeta(2n), etc.). We are disappointed that some of the most interesting paths, especially Euler's proposed approach to the four squares theorem, lead to elliptic functions and will not be followed up in this book. Chapter 4 on Lagrange introduces a topic that in fact will be studied thoroughly in this book: quadratic forms, which is a powerful and systematic approach to many of the most important theorems of Fermat. We also study continued fractions, since the continued fraction algorithm applied to sqrt(d) produces the solutions of the Pell equation x^2-dy^2=1. Fermat-style number theory begs for the law of quadratic reciprocity as a unifying tool, as Legandre saw (chapter 5), but proper proofs of this theorem was only given by Gauss (chapter 6). We study his fourth and sixth proofs; these are based on "Gauss sums", which Gauss ran into in the context of cyclotomy. We also look at Gauss's theory of quadratic forms and its more modern sister theory, the algebraic approach to quadratic integers in terms of ideal theory. Then there is a short chapter 7 on Fourier, who didn't have much to do with number theory per se, but whose Fourier series immediately pays off, for instance by providing a quick path to Euler's result zeta(2)=pi^2/6. Then there is a long chapter 8 on Dirichlet's analytic number theory. First we tie up a loose end from the Gauss chapter, the evaluation of Gauss sums. Then we look at Dirichlet's proof that there are infinitely many primes in essentially any arithmetic progression, which shines especially bright in a historical treatment since it involves zeta function ideas going back to Euler as well as providing the necessary machinery for filling a big gap in Gauss's theory of quadratic forms: the determination of the class number. Actually, the proof of the nonvanishing of the L-series at 1 follows Landau rather than Dirichlet's proof based on "virtuoser Rechenkunst". In chapters 9 and 10 we study quadratic forms in n variables. Minkowski's reformulation of the theory of quadratic forms in terms of the geometry of lattices leads to swift (almost "Rechenkunst"-free) proofs of great classical theorems like the four squares theorem, while at the same time pointing the way to a systematic study of groups of linear transformations, e.g. Haar measure on SL(n,R) and such.Von Fermat bis Minkowski: Eine Vorlesung über Zahlentheorie und ihre Entwicklung (German Edition) Overview

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