Commutative Algebra 1

From the Preface: "We have preferred to write a self-contained book which could be used in a basic graduate course of modern algebra. It is also with an eye to the student that we have tried to give full and detailed explanations in the proofs... We have also tried, this time with an eye to both the student and the mature mathematician, to give a many-sided treatment of our topics, not hesitating to offer several proofs of one and the same result when we thought that something might be learned, as to methods, from each of the proofs." Title Preface Contents I. Introductory concepts § 1. Binary operations § 2. Groups § 3. Subgroups § 4. Abelian groups § 5. Rings § 6. Rings with identity § 7. Powers and multiples § 8. Fields § 9. Subrings and subfields § 10. Transformations and mappings § 11. Group homomorphisms § 12. Ring homomorphisms § 13. Identification of rings § 14. Unique factorization domains § 15. Euclidean domains § 16. Polynomials in one indeterminate § 17. Polynomial rings § 18. Polynomials in several indeterminates § 19. Quotient fields and total quotient rings § 20. Quotient rings with respect to multiplicative systems § 21. Vector spaces II. Elements of field theory § 1. Field extensions § 2. Algebraic quantities § 3. Algebraic extensions § 4. The characteristic of a field § 5. Separable and inseparable algebraic extension § 6. Splitting fields and normal extensions § 7. The fundamental theorem of Galois theory § 8. Galois fields § 9. The theorem of the primitive element § 10. Field polynomials. Norms and traces § 11. The discriminant § 12. Transcendental extensions § 13. Separably generated fields of algebraic functions § 14. Algebraically closed fields § 15. Linear disjointness and separability § 16. Order of inseparability of a field of algebraic functions § 17. Derivations III. Ideals and modules § 1. Ideals and modules § 2. Operations on submodules § 3. Operator homomorphisms and difference modules § 4. The isomorphism theorems § 5. Ring homomorphisms and residue class rings § 6. The order of a subset of a module § 7. Operations on ideals § 8. Prime and maximal ideals § 9. Primary ideals § 10. Finiteness conditions § 11. Composition series § 12. Direct sums § 12bis. Infinite direct sums § 13. Comaximal ideals and direct sums of ideals § 14. Tensor products of rings § 15. Free joins of integral domains (or of fields) IV. Noetherian rings § 1. Definitions. The Hilbert basis theorem § 2. Rings with descending chain condition § 3. Primary rings § 3bis. Alternative method for studying the rings with d.c.c. § 4. The Lasker-Noether decomposition theorem § 5. Uniqueness theorems § 6. Application to zero-divisors and nilpotent elements § 7. Application to the intersection of the powers of an ideal § 8. Extended and contracted ideals § 9. Quotient rings § 10. Relations between ideals in R and ideals in R_M § 11. Examples and applications of quotient rings § 12. Symbolic powers § 13. Length of an ideal § 14. Prime ideals in noetherian rings § 15. Principal ideal rings § 16. Irreducible ideals Appendix: Primary representation in Noetherian modules V. Dedekind domains, classical ideal theory § 1. Integral elements § 2. Integrally dependent rings § 3. Integrally closed rings § 4. Finiteness theorems § 5. The conductor of an integral closure § 6. Characterizations of Dedekind domains § 7. Further properties of Dedekind domains § 8. Extensions of Dedekind domains § 9. Decomposition of prime ideals in extensions of Dedekind domains § 10. Decomposition group, inertia group, and ramification groups § 11. Different and discriminant § 12. Application to quadratic fields and cyclotomic fields § 13. A theorem of Kummer Index of notations Index of definitions Vol. 2. Valuation Theory -- Polynomial And Power Series Rings -- Local Algebra -- Appendix -- Index Of Definitions. Oscar Zariski, Pierre Samuel, With The Cooperation Of I. S. Cohen. Reprint Of The Ed. Published By Van Nostrand, Princeton, N.j., In Series: The University Series In Higher Mathematics. Includes Indexes.