Introduction
1. Cyclohexane, cryptography, codes, and computer algebra
Part I. Euclid:
2. Fundamental algorithms
3. The Euclidean Algorithm
4. Applications of the Euclidean Algorithm
5. Modular algorithms and interpolation
6. The resultant and gcd computation
7. Application: decoding BCH codes
Part II. Newton:
8. Fast multiplication
9. Newton iteration
10. Fast polynomial evaluation and interpolation
11. Fast Euclidean Algorithm
12. Fast linear algebra
13. Fourier Transform and image compression
Part III. Gauß:
14. Factoring polynomials over finite fields
15. Hensel lifting and factoring polynomials
16. Short vectors in lattices
17. Applications of basis reduction
Part IV. Fermat:
18. Primality testing
19. Factoring integers
20. Application: public key cryptography
Part V. Hilbert:
21. Gröbner bases
22. Symbolic integration
23. Symbolic summation
24. Applications
Appendix:
25. Fundamental concepts
Sources of illustrations
Sources of quotations
List of algorithms
List of figures and tables
References
List of notation
Index.

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Detailed algorithms with analysis, and complete proofs (none left to the reader)
Student-friendly with gentle introductions and overviews
Includes many illustrations and applications and much historical background
Solutions to selected exercises are available from the book's webpage