© 2017

From Natural Numbers to Quaternions


Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Jürg Kramer, Anna-Maria von Pippich
    Pages 9-43
  3. Jürg Kramer, Anna-Maria von Pippich
    Pages 45-91
  4. Jürg Kramer, Anna-Maria von Pippich
    Pages 93-139
  5. Jürg Kramer, Anna-Maria von Pippich
    Pages 141-182
  6. Jürg Kramer, Anna-Maria von Pippich
    Pages 183-217
  7. Jürg Kramer, Anna-Maria von Pippich
    Pages 219-246
  8. Back Matter
    Pages 247-285

About this book


This textbook offers an invitation to modern algebra through number systems of increasing complexity, beginning with the natural numbers and culminating with Hamilton's quaternions.

Along the way, the authors carefully develop the necessary concepts and methods from abstract algebra: monoids, groups, rings, fields, and skew fields. Each chapter ends with an appendix discussing related topics from algebra and number theory, including recent developments reflecting the relevance of the material to current research.

The present volume is intended for undergraduate courses in abstract algebra or elementary number theory. The inclusion of exercises with solutions also makes it suitable for self-study and accessible to anyone with an interest in modern algebra and number theory.


MSC (2010): 08–01, 11–01, 12–01, 20–01 integers construction rational numbers construction real numbers construction complex numbers construction Hamiltonian quaternions construction group theory elements ring theory elements complex numbers algebraicity proof transcendence Euler number e Associative Rings and Algebras Commutative Rings and Algebras Field Theory and Polynomials Group Theory and Generalizations Number Theory

Authors and affiliations

  1. 1.Department of MathematicsHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany

About the authors

Jürg Kramer is Professor of Mathematics at the Humboldt-Universität zu Berlin, Germany. His research focuses on arithmetic geometry, in particular on Arakelov geometry, and the theory of modular and automorphic forms. He is also interested in questions about the teaching of mathematics at university level.

Anna-Maria von Pippich is Junior Professor of Algebra and Number Theory at the Technische Universität Darmstadt, Germany. She is working in number theory, in particular in the theory of automorphic forms, and Arakelov geometry.

Bibliographic information

  • Book Title From Natural Numbers to Quaternions
  • Authors Jürg Kramer
    Anna-Maria von Pippich
  • Series Title Springer Undergraduate Mathematics Series
  • Series Abbreviated Title SUMS
  • DOI
  • Copyright Information Springer International Publishing AG 2017
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-69427-6
  • eBook ISBN 978-3-319-69429-0
  • Series ISSN 1615-2085
  • Series E-ISSN 2197-4144
  • Edition Number 1
  • Number of Pages XVIII, 277
  • Number of Illustrations 4 b/w illustrations, 6 illustrations in colour
  • Additional Information Original German edition published by Springer Spektrum, Wiesbaden, 2013
  • Topics Algebra
    Number Theory
  • Buy this book on publisher's site