Computer Algebra

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.

Table of Contents

Chapter 1 Introduction to Computer Algebra

Chapter 2 Programming in Computer Algebra Systems

Chapter 3 Number Systems and Integer Arithmetic

Chapter 4 Modular Arithmetic

Chapter 5 Coding Theory and Cryptography

Chapter 6 Polynomial Arithmetic

Chapter 7 Algebraic Numbers

Chapter 8 Factorization in Polynomial Rings

Chapter 9 Simplification and Normal Forms

Chapter 10 Power Series

Chapter 11 Algorithmic Summation

Chapter 12 Algorithmic Integration

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).