Mathematics for Computer Algebra

1 Elementary Arithmetics.- 1. Representation of an integer in basis B1.- 2. Addition.- 3. Subtraction.- 4. Multiplication.- 5. Euclidean division.- 6. The cost of multiplication and division.- 7. How to compute powers.- 8. The g.c.d..- 9. The group G (n).- 10. The Chinese remainder theorem.- 11. The prime numbers.- 2 Number Theory, Complements.- 1. Study of the group G(n).- 2. Tests of primality.- 3. Factorization of rational integers.- 3 Polynomials, Algebraic Study.- 1. Definitions and elementary properties.- 2. Euclidean division.- 3. The Chinese remainder theorem.- 4. Factorization.- 5. Polynomial functions.- 6. The resultant.- 7. Companion matrix.- 8. Linear recursive sequences.- 4 Polynomials with complex coefficients.- 1. The theorem of d'Alembert.- 2. Estimates of the roots.- 3. The measure of a polynomial.- 4. Bounds for size of the factors of a polynomial.- 5. The distribution of the roots of a polynomial.- 6. Separation of the roots of a polynomial.- 5 Polynomials with real coefficients.- 1. Polynomials irreducible over ?.- 2. The theorem of Rolle.- 3. Estimates of real roots.- 4. The number of zeros of a polynomial in a real interval.- 5. Equations whose roots have a negative real part.- 6/Polynomials over finite fields.- 1. Finite fields.- 2. Statistics on Hq[X].- 3. Factorization into a product of squarefree polynomials.- 4. Factorization of the polynomials over a finite field.- 5. Search for the roots of a polynomial in a finite field.- 7 Polynomials with integer coefficients.- 1. Principles of the algorithms of factorization.- 2. The choice of the prime modulus.- 3. Refining the factorization.- 4. Berlekamp's method of factorization.- 5. The algorithm L3.- 6. Factors of polynomials and lattices.- 7. The algorithm of factorization.- Index of Names.