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.

A selective classified bibliography of symbolic computation in some areas of chemistry is provided together with some examples of computer algebra algorithms and techniques to facilitate future joint work of chemists and computer scientists.

Parallel algorithms to compute the determinant and characteristic polynomial of matrices and the gcd of polynomials are presented. The rank of matrices and solutions of arbitrary systems of linear equations are computed by parallel Las Vegas algorithms. All algorithms work over arbitrary fields. They run in parallel time O(log ~ n) (where n is the number of inputs) and use a polynomial number of processors.

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Gauss periods yield (self-dual) normal bases in finite fields, and these normal bases can be used to implement arithmetic efficiently. It is shown that for a small prime power q and infinitely many integers n, multiplication in a normal basis of F q n over Fq can be computed with O(n log n loglog n), division with O(n log 2 n loglog n) operations in Fq, and exponentiation of an arbitrary element in F q n with O(n 2 loglog n) operations in Fq. We also prove that using a polynomial basis exponentiation in F 2 n can be done with the same number of operations in F 2 for all n. The previous best estimates were O(n 2 ) for multiplication in a normal basis, and O(n 2 log n log log n) for exponentiation in a polynomial basis.