From Fermat to Minkowski: Lectures on the Theory of Numbers and Its Historical Development (Undergraduate Texts in Mathematics) 1st Edition by Winfried Scharlau (PDF)

Description

This book arose from a course of lectures given by the first author during the winter term 1977/1978 at the University of Münster (West Germany). The course was primarily addressed to future high school teachers of mathematics; it was not meant as a systematic introduction to number theory but rather as a historically motivated invitation to the subject, designed to interest the audience in number-theoretical questions and developments. This is also the objective of this book, which is certainly not meant to replace any of the existing excellent texts in number theory. Our selection of topics and examples tries to show how, in the historical development, the investigation of obvious or natural questions has led to more and more comprehensive and profound theories, how again and again, surprising connections between seemingly unrelated problems were discovered, and how the introduction of new methods and concepts led to the solution of hitherto unassailable questions. All this means that we do not present the student with polished proofs (which in turn are the fruit of a long historical development); rather, we try to show how these theorems are the necessary consequences of natural questions. Two examples might illustrate our objectives.

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⭐ミュンスター大学で将来高校の数学の先生になる学生に行った講義がもとになっていて（序文より）、起源から現代までの数論の発展の歴史が一望できる内容で、各時代の中心的な数論の研究者の、肖像写真と生い立ちや同時代の数学者、研究内容の紹介という感じの構成になっています。「数学の歴史」的なタイトルの本から数論関係の部分を集めて、ちょっと詳しくした、と思えばいいかもしれません。フェルマーの章では、数々の有名な定理とともに「方程式、ｘの３乗＝ｙの２乗＋２、の唯一の自然数の解はｘ＝３、ｙ＝５である」というあまり見かけない面白い定理も出ています。他に目についた言葉を拾っていくと、オイラーの章、初等関数の級数展開、ベルヌイ数、ｎの２乗分の１の無限和＝６分のπの２乗、ラグランジュ、２元２次形式で表される整数、連分数展開、ルジャンドル、平方剰余の相互法則、ガウス、正１７角形、ガロア理論、代数学の基本定理、ζ（ゼータ）関数・・・など、どこかで目にしている定理が多いですが、日本語訳を知らない言葉や、読めない名前の数学者もたくさんありました。日本語の類書もありますし、特にどういう人におすすめということではありませんが、手軽に入手可能な面白い読み物、と言えるのではないでしょうか。

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