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Lectures on algebra. Volume I / S.S. Abhyankar.

By: Abhyankar, Shreeram ShankarMaterial type: TextTextPublication details: Singapore ; Hackensack, N.J. : World Scientific, ©2006. Description: 1 online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9812773444; 9789812773449Subject(s): Algebra | Algebra, Abstract | Algebras, Linear | MATHEMATICS -- Algebra -- Intermediate | Algebra | Algebra, Abstract | Algebras, LinearGenre/Form: Electronic books. | Electronic books. DDC classification: 512 LOC classification: QA154.3Online resources: Click here to access online
Contents:
Lecture L1. Quadratic equations. 1. Word problems. 2. Sets and maps. 3. Groups and fields. 4. Rings and ideals. 5. Modules and vector spaces. 6. Polynomials and rational functions. 7. Euclidean domains and principal ideal domains. 8. Root fields and splitting fields. 9. Advice to the reader. 10. Definitions and remarks. 11. Examples and exercises. 12. Notes. 13. Concluding note -- Lecture L2. Curves and surfaces. 1. Multivariable word problems. 2. Power series and meromorphic series. 3. Valuations. 4. Advice to the reader. 5. Zorn's Lemma and well ordering. 6. Utilitarian summary. 7. Definitions and exercises. 8. Notes. 9. Concluding note.
Lecture L3. Tangents and polars. 1. Simple groups. 2. Quadrics. 3. Hypersurfaces. 4. Homogeneous coordinates. 5. Singularities. 6. Hensel's Lemma and Newton's theorem. 7. Integral dependence. 8. Unique factorization domains. 9. Remarks. 10. Advice to the reader. 11. Hensel and Weierstrass. 12. Definitions and exercises. 13. Notes. 14. Concluding note -- Lecture L4. Varieties and models. 1. Resultants and discriminants. 2. Varieties. 3. Noetherian rings. 4. Advice to the reader. 5. Ideals and modules. 6. Primary decomposition. 7. Localization. 8. Affine varieties. 9. Models. 10. Examples and exercises. 11. Problems. 12. Remarks. 13. Definitions and exercises. 14. Notes. 15. Concluding note -- Lecture L5. Projective varieties. 1. Direct sums of modules. 2. Grades rings and homogeneous ideals. 3. Ideal theory in graded rings. 4. Advice to the reader. 5. More about ideals and modules -- Q1. Nilpotents and zerodivisors in Noetherian rings.
Q2. Faithful modules and Noetherian conditions -- Q3. Jacobson radical, Zariski ring, and Nakayama Lemma -- Q4. Krull intersection theorem and Artin-Rees Lemma -- Q5. Nagata's principle of idealization -- Q6. Cohen's and Eakin's Noetherian theorems -- Q7. Principal ideal theorems -- Q8. Relative independence and analytic independence -- Q9. Going up and going down theorems -- Q10. Normalization theorem and regular polynomials -- Q11. Nilradical, Jacobson Spectrum, and Jacobson Ring -- Q12. Catenarian Rings and dimension formula -- Q13. Associated graded rings and leading ideals -- Q14. Completely normal domains -- Q15. Regular sequences and Cohen-Macaulay rings -- Q16. Complete intersections and Gorenstein Rings -- Q17. Projective resolutions of finite modules -- Q18. Direct sums of algebras, reduced rings, and PIRs -- Q19. Invertible ideals, conditions for normality, and DVRs -- Q20. Dedekind domains and Chinese remainder theorem.
Q21. Real ranks of valuations and segment completions -- Q22. Specializations and compositions of valuations -- Q23. UFD property of regular local domains -- Q24. Graded modules and Hilbert polynomials -- Q25. Hilbert polynomial of a hypersurfaces -- Q26. Homogeneous submodules of graded modules -- Q27. Homogeneous normalization -- Q28. Alternating sum of lengths -- Q29. Linear disjointness and intersection of varieties -- Q30. Syzygies and homogeneous resolutions -- Q31. Projective modules over polynomial rings -- Q32. Separable extensions and primitive elements -- Q33. Restricted domains and projective normalization -- Q34. Basic projective algebraic geometry -- Q. 35. Simplifying singularities by blowups.
Summary: This book is a timely survey of much of the algebra developed during the last several centuries including its applications to algebraic geometry and its potential use in geometric modeling. The present volume makes an ideal textbook for an abstract algebra course, while the forthcoming sequel, Lectures on Algebra II, will serve as a textbook for a linear algebra course. The author's fondness for algebraic geometry shows up in both volumes, and his recent preoccupation with the applications of group theory to the calculation of Galois groups is evident in the second volume which contains more.
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Includes bibliographical references (pages 689-690) and index.

Lecture L1. Quadratic equations. 1. Word problems. 2. Sets and maps. 3. Groups and fields. 4. Rings and ideals. 5. Modules and vector spaces. 6. Polynomials and rational functions. 7. Euclidean domains and principal ideal domains. 8. Root fields and splitting fields. 9. Advice to the reader. 10. Definitions and remarks. 11. Examples and exercises. 12. Notes. 13. Concluding note -- Lecture L2. Curves and surfaces. 1. Multivariable word problems. 2. Power series and meromorphic series. 3. Valuations. 4. Advice to the reader. 5. Zorn's Lemma and well ordering. 6. Utilitarian summary. 7. Definitions and exercises. 8. Notes. 9. Concluding note.

Lecture L3. Tangents and polars. 1. Simple groups. 2. Quadrics. 3. Hypersurfaces. 4. Homogeneous coordinates. 5. Singularities. 6. Hensel's Lemma and Newton's theorem. 7. Integral dependence. 8. Unique factorization domains. 9. Remarks. 10. Advice to the reader. 11. Hensel and Weierstrass. 12. Definitions and exercises. 13. Notes. 14. Concluding note -- Lecture L4. Varieties and models. 1. Resultants and discriminants. 2. Varieties. 3. Noetherian rings. 4. Advice to the reader. 5. Ideals and modules. 6. Primary decomposition. 7. Localization. 8. Affine varieties. 9. Models. 10. Examples and exercises. 11. Problems. 12. Remarks. 13. Definitions and exercises. 14. Notes. 15. Concluding note -- Lecture L5. Projective varieties. 1. Direct sums of modules. 2. Grades rings and homogeneous ideals. 3. Ideal theory in graded rings. 4. Advice to the reader. 5. More about ideals and modules -- Q1. Nilpotents and zerodivisors in Noetherian rings.

Q2. Faithful modules and Noetherian conditions -- Q3. Jacobson radical, Zariski ring, and Nakayama Lemma -- Q4. Krull intersection theorem and Artin-Rees Lemma -- Q5. Nagata's principle of idealization -- Q6. Cohen's and Eakin's Noetherian theorems -- Q7. Principal ideal theorems -- Q8. Relative independence and analytic independence -- Q9. Going up and going down theorems -- Q10. Normalization theorem and regular polynomials -- Q11. Nilradical, Jacobson Spectrum, and Jacobson Ring -- Q12. Catenarian Rings and dimension formula -- Q13. Associated graded rings and leading ideals -- Q14. Completely normal domains -- Q15. Regular sequences and Cohen-Macaulay rings -- Q16. Complete intersections and Gorenstein Rings -- Q17. Projective resolutions of finite modules -- Q18. Direct sums of algebras, reduced rings, and PIRs -- Q19. Invertible ideals, conditions for normality, and DVRs -- Q20. Dedekind domains and Chinese remainder theorem.

Q21. Real ranks of valuations and segment completions -- Q22. Specializations and compositions of valuations -- Q23. UFD property of regular local domains -- Q24. Graded modules and Hilbert polynomials -- Q25. Hilbert polynomial of a hypersurfaces -- Q26. Homogeneous submodules of graded modules -- Q27. Homogeneous normalization -- Q28. Alternating sum of lengths -- Q29. Linear disjointness and intersection of varieties -- Q30. Syzygies and homogeneous resolutions -- Q31. Projective modules over polynomial rings -- Q32. Separable extensions and primitive elements -- Q33. Restricted domains and projective normalization -- Q34. Basic projective algebraic geometry -- Q. 35. Simplifying singularities by blowups.

Lecture L6. Pause and refresh. 1. Summary of Lecture L1 on quadratic equations. 2. Summary of Lecture L2 on curves and surfaces. 3. Summary of Lecture L3 on tangents and polars. 4. Summary of Lecture L4 on varieties and models. 5. Summary of Lecture L5 on projective varieties. 6. Definitions and exercises.

This book is a timely survey of much of the algebra developed during the last several centuries including its applications to algebraic geometry and its potential use in geometric modeling. The present volume makes an ideal textbook for an abstract algebra course, while the forthcoming sequel, Lectures on Algebra II, will serve as a textbook for a linear algebra course. The author's fondness for algebraic geometry shows up in both volumes, and his recent preoccupation with the applications of group theory to the calculation of Galois groups is evident in the second volume which contains more.

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