Modern computer algebra 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 crytopgraphy 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. 1. Cyclohexane cryptography codes and computer algebra 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 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 14. Factoring polynomials over finite fields 15. Hensel lifting and factoring polynomials 16. Short vectors in lattices 17. Applications of basis reduction 18. Primality testing 19. Factoring integers 20. Application: public key cryptography 21. Gröbner bases 22. Symbolic integration 23. Symbolic summation 24. Applications 25. Fundamental concepts