1 Introduction. - 1. 1 Prologue: Algebra and Algorithms. - 1. 2 Motivations. - 1. 3 Algorithmic Notations. - 1. 4 Epilogue. - Bibliographic Notes. - 2 Algebraic Preliminaries. - 2. 1 Introduction to Rings and Ideals. - 2. 2 Polynomial Rings. - 2. 3 Gröbner Bases. - 2. 4 Modules and Syzygies. - 2. 5 S-Polynomials. - Problems. - Solutions to Selected Problems. - Bibliographic Notes. - 3 Computational Ideal Theory. - 3. 1 Introduction. - 3. 2 Strongly Computable Ring. - 3. 3 Head Reductions and Gröbner Bases. - 3. 4 Detachability Computation. - 3. 5 Syzygy Computation. - 3. 6 Hilbert's Basis Theorem: Revisited. - 3. 7 Applications of Gröbner Bases Algorithms. - Problems. - Solutions to Selected Problems. - Bibliographic Notes. - 4 Solving Systems of Polynomial Equations. - 4. 1 Introduction. - 4. 2 Triangular Set. - 4. 3 Some Algebraic Geometry. - 4. 4 Finding the Zeros. - Problems. - Solutions to Selected Problems. - Bibliographic Notes. - 5 Characteristic Sets. - 5. 1 Introduction. - 5. 2 Pseudodivision and Successive Pseudodivision. - 5. 3 Characteristic Sets. - 5. 4 Properties of Characteristic Sets. - 5. 5 Wu-Ritt Process. - 5. 6 Computation. - 5. 7 Geometric Theorem Proving. - Problems. - Solutions to Selected Problems. - Bibliographic Notes. - 6 An Algebraic Interlude. - 6. 1 Introduction. - 6. 2 Unique Factorization Domain. - 6. 3 Principal Ideal Domain. - 6. 4 Euclidean Domain. - 6. 5 Gauss Lemma. - 6. 6 Strongly Computable Euclidean Domains. - Problems. - Solutions to Selected Problems. - Bibliographic Notes. - 7 Resultants and Subresultants. - 7. 1 Introduction. - 7. 2 Resultants. - 7. 3 Homomorphisms and Resultants. - 7. 4 Repeated Factors in Polynomials and Discriminants. - 7. 5 Determinant Polynomial. - 7. 6 Polynomial Remainder Sequences. - 7. 7 Subresultants. - 7. 8 Homomorphisms and Subresultants. - 7. 9 Subresultant Chain. - 7. 10 Subresultant ChainTheorem. - Problems. - Solutions to Selected Problems. - Bibliographic Notes. - 8 Real Algebra. - 8. 1 Introduction. - 8. 2 Real Closed Fields. - 8. 3 Bounds on the Roots. - 8. 4 Sturm's Theorem. - 8. 5 Real Algebraic Numbers. - 8. 6 Real Geometry. - Problems. - Solutions to Selected Problems. - Bibliographic Notes. - Appendix A: Matrix Algebra. - A. 1 Matrices. - A. 2 Determinant. - A. 3 Linear Equations. Language: English