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Computer Algebra

An Algorithm-Oriented Introduction

  • Textbook
  • © 2021

Overview

Authors:
  1. Wolfram Koepf

    1. Institut für Mathematik, Universität Kassel, Kassel, Germany

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  • Balances theoretical and practical aspects with an implementation-oriented approach
  • Guides readers through numerous hands-on tutorials to build practical skills and algorithmic thinking
  • Supports three computer algebra systems with downloadable resources in Mathematica, Maple, and Maxima

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Table of contents (12 chapters)

  1. Front Matter

    Pages I-XII
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  2. Introduction to Computer Algebra

    • Wolfram Koepf
    Pages 1-20

  3. Programming in Computer Algebra Systems

    • Wolfram Koepf
    Pages 21-38

  4. Number Systems and Integer Arithmetic

    • Wolfram Koepf
    Pages 39-64

  5. Modular Arithmetic

    • Wolfram Koepf
    Pages 65-89

  6. Coding Theory and Cryptography

    • Wolfram Koepf
    Pages 91-112

  7. Polynomial Arithmetic

    • Wolfram Koepf
    Pages 113-150

  8. Algebraic Numbers

    • Wolfram Koepf
    Pages 151-191

  9. Factorization in Polynomial Rings

    • Wolfram Koepf
    Pages 193-219

  10. Simplification and Normal Forms

    • Wolfram Koepf
    Pages 221-234

  11. Power Series

    • Wolfram Koepf
    Pages 235-288

  12. Algorithmic Summation

    • Wolfram Koepf
    Pages 289-328

  13. Algorithmic Integration

    • Wolfram Koepf
    Pages 329-360

  14. Back Matter

    Pages 361-384
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Keywords

About this book

This textbook offers an algorithmic introduction to the field of computer algebra. A leading expert in the field, the author guides readers through numerous hands-on tutorials designed to build practical skills and algorithmic thinking. This implementation-oriented approach equips readers with versatile tools that can be used to enhance studies in mathematical theory, applications, or teaching. Presented using Mathematica code, the book is fully supported by downloadable sessions in Mathematica, Maple, and Maxima.

Opening with an introduction to computer algebra systems and the basics of programming mathematical algorithms, the book goes on to explore integer arithmetic. A chapter on modular arithmetic completes the number-theoretic foundations, which are then applied to coding theory and cryptography. From here, the focus shifts to polynomial arithmetic and algebraic numbers, with modern algorithms allowing the efficient factorization of polynomials. The final chapters offer extensions into more advanced topics: simplification and normal forms, power series, summation formulas, and integration.

Computer Algebra is an indispensable resource for mathematics and computer science students new to the field. Numerous examples illustrate algorithms and their implementation throughout, with online support materials to encourage hands-on exploration. Prerequisites are minimal, with only a knowledge of calculus and linear algebra assumed. In addition to classroom use, the elementary approach and detailed index make this book an ideal reference for algorithms in computer algebra.

Reviews

“Strong interplay between the abstract exposition, which includes the relevant theorems as well as their proofs, and the practical utilization of those concepts in Mathematica is certainly a remarkable feature of this textbook. … Overall, the book is very well written and the approach to provide examples as actual Mathematica sessions is commendable.” (Andreas Maletti, zbMATH 1484.68004, 2022)

Authors and Affiliations

  • Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

About the author

Wolfram Koepf is Professor of Mathematics (retired) at the University of Kassel. He is a leading expert in computer algebra. Between 2002 and 2010 he was the Chairman of the Fachgruppe Computeralgebra, the largest computer algebra group worldwide. He has organized numerous conferences and workshops around the world, including the International Symposium on Symbolic and Algebraic Computation (ISSAC) 2010 as general chair, and a cooperation with the African Institute of Mathematical Sciences Cameroon in 2017 and 2018. In addition to his prolific research and organizational activities, he has a strong interest in education and teacher preparation. He is also a member of the executive committee of the German Mathematical Union (DMV).

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