From Natural Numbers to Quaternions

  • Jürg Kramer
  • Anna-Maria von Pippich

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

  • Jürg Kramer
    • 1
  • Anna-Maria von Pippich
    • 2
  1. 1.Department of MathematicsHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany

Bibliographic information

  • DOI
  • Copyright Information Springer International Publishing AG 2017
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-69427-6
  • Online ISBN 978-3-319-69429-0
  • Series Print ISSN 1615-2085
  • Series Online ISSN 2197-4144
  • Buy this book on publisher's site