Hurwitz's lectures on the number theory of quaternions / Nicola Oswald, Jörn Steuding.

Format:

Book

Author/Creator:

Oswald, Nicola, author.

Steuding, Jörn, author.

Series:

Heritage of European mathematics

Language:

English

Subjects (All):

Quaternions.

Number theory.

Hurwitz, Adolf, 1859-1919. Works--Selections.

Hurwitz, Adolf.

Physical Description:

xvii, 293 pages : illustrations ; 25 cm.

Place of Publication:

Berlin, Germany : EMS Press, an imprint of the European Mathematical Society, [2023]

Summary:

Quaternions are non-commutative generalizations of the complex numbers, invented by William Rowan Hamilton in 1843. Their number-theoretical aspects were first investigated by Rudolf Lipschitz in the 1880s, and, in a streamlined form, by Adolf Hurwitz in 1896. This book contains an English translation of Hurwitz's 1919 textbook on this topic as well as his famous 1-2-3-4 theorem on composition algebras. In addition, the reader can find commentaries that shed historical light on the development of this number theory of quaternions, for example, the classical preparatory works of Fermat, Euler, Lagrange and Gauss, to name but a few, the different notions of quaternion integers in the works of Lipschitz and Hurwitz, analogies to the theory of algebraic numbers, and the further development (including Dickson's work in particular). The authors have implemented parts of the book in stand-alone courses, and they believe that the present book can also complement a course on algebraic number theory (with respect to a noncommutative extension of the rational numbers).

Contents:

Introduction

Lectures on the number theory of quaternions

Preface

The quaternions and how to compute with them

The field of quaternions and their permutations and inversions

The field R and its permutations

The integer quaternions

The permutations of integer quaternions

Greatest common divisor and quaternion ideals

Even and odd quaternions. Associated and primary quaternions

The integer quaternions modulo an odd number

The prime quaternions

The factorization theorem

The representations of a positive integer as a sum of four squares

A problem due to Euler

Notes and addendum

Two Arithmetics

A view into Hurwitz’s Mathematische Tagebücher

On the composition of quadratic forms

Abstraction and generalization

Epilogue

Appendix A: Theses

Appendix B: Elementary number theory in a nutshell.

Notes:

Includes bibliographical references and index.

ISBN:

9783985470112

3985470111

OCLC:

1395538043

Publisher Number:

99996552869