Lectures on algebra : Volume 1

Includes index

Includes bibliography (p. 689-690) and references

Lecture LI: 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 Exercises8: 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.6.1: Primary Decomposition for Modules7: Localization; 7.1: Localization at a Prime Ideal; 8: Affine Varieties; 8.1: Spectral Affine Space; 8.2: Modelic Spec and Modelic Affine Space; 8.3: Simple Points and Regular Local Rings; 9: Models; 9.1: Modelic Proj and Modelic Projective Space; 9.2: Modelic Blowup; 9.3: Blowup of Singularities; 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: Graded Rings and Homogeneous Ideals; 3: Ideal Theory in Graded Rings.4: Advice to the Reader5: More about Ideals and Modules; (Ql) 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; (Qll) 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; (Q18.1) Orthogonal Idempotents and Ideals in a Direct Sum; (Q18.2) Localizations of Direct Sums; (Q18.3) Comaximal Ideals and Ideal Theoretic Direct Sums; (Q18.4) SPIRs = Special Principal Ideal Rings; (Q19) Invertible Ideals Conditions for Normality and DVRs.

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.