Intro
Preface
Contents
Chapter 1 Introduction to Computer Algebra
1.1 Capabilities of Computer Algebra Systems
1.2 Additional Remarks
1.3 Exercises
Chapter 2 Programming in Computer Algebra Systems
2.1 Internal Representation of Expressions
2.2 Pattern Matching
2.3 Control Structures
2.4 Recursion and Iteration
2.5 Remember Programming
2.6 Divide-and-Conquer Programming
2.7 Programming through Pattern Matching
2.8 Additional Remarks
2.9 Exercises
Chapter 3 Number Systems and Integer Arithmetic
3.1 Number Systems

3.2 Integer Arithmetic: Addition and Multiplication
3.3 Integer Arithmetic: Division with Remainder
3.4 The Extended Euclidean Algorithm
3.5 Unique Factorization
3.6 Rational Arithmetic
3.7 Additional Remarks
3.8 Exercises
Chapter 4 Modular Arithmetic
4.1 Residue Class Rings
4.2 Modulare Square Roots
4.3 Chinese Remainder Theorem
4.4 Fermat's Little Theorem
4.5 Modular Logarithms
4.6 Pseudoprimes
4.7 Additional Remarks
4.8 Exercises
Chapter 5 Coding Theory and Cryptography
5.1 Basic Concepts of Coding Theory
5.2 Prefix Codes

5.3 Check Digit Systems
5.4 Error Correcting Codes
5.5 Asymmetric Ciphers
5.6 Additional Remarks
5.7 Exercises
Chapter 6 Polynomial Arithmetic
6.1 Polynomial Rings
6.2 Multiplication: The Karatsuba Algorithm
6.3 Fast Multiplication with FFT
6.4 Division with Remainder
6.5 Polynomial Interpolation
6.6 The Extended Euclidean Algorithm
6.7 Unique Factorization
6.8 Squarefree Factorization
6.9 Rational Functions
6.10 Additional Remarks
6.11 Exercises
Chapter 7 Algebraic Numbers
7.1 Polynomial Quotient Rings
7.2 Chinese Remainder Theorem

7.3 Algebraic Numbers
7.4 Finite Fields
7.5 Resultants
7.6 Polynomial Systems of Equations
7.7 Additional Remarks
7.8 Exercises
Chapter 8 Factorization in Polynomial Rings
8.1 Preliminary Considerations
8.2 Efficient Factorization in Zp[x]
8.3 Squarefree Factorization of Polynomials over Finite Fields
8.4 Efficient Factorization in Q[x]
8.5 Hensel Lifting
8.6 Multivariate Factorization
8.7 Additional Remarks
8.8 Exercises
Chapter 9 Simplification and Normal Forms
9.1 Normal Forms and Canonical Forms

9.2 Normal Forms and Canonical Forms for Polynomials
9.3 Normal Forms for Rational Functions
9.4 Normal Forms for Trigonometric Polynomials
9.5 Additional Remarks
9.6 Exercises
Chapter 10 Power Series
10.1 Formal Power Series
10.2 Taylor Polynomials
10.3 Computation of Formal Power Series
10.3.1 Holonomic Differential Equations
10.3.2 Holonomic Recurrence Equations
10.3.3 Hypergeometric Functions
10.3.4 Efficient Computation of Taylor Polynomials of Holonomic Functions
10.4 Algebraic Functions
10.5 Implicit Functions
10.6 Additional Remarks