**Graduate Studies in Mathematics**

Volume: 73;
2006;
438 pp;

MSC: Primary 28;
Secondary 26; 31; 42; 46; 49; 81

**Electronic ISBN: 978-1-4704-1152-7
Product Code: GSM/73.E**

List Price: $71.00

AMS Member Price: $56.80

MAA Member Price: $63.90

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#### Supplemental Materials

# Graduate Algebra: Commutative View

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*Louis Halle Rowen*

This book is an expanded text for a graduate course in
commutative algebra, focusing on the algebraic underpinnings of algebraic
geometry and of number theory. Accordingly, the theory of affine algebras is
featured, treated both directly and via the theory of Noetherian and Artinian
modules, and the theory of graded algebras is included to provide the
foundation for projective varieties. Major topics include the theory of modules
over a principal ideal domain, and its applications to matrix theory (including
the Jordan decomposition), the Galois theory of field extensions, transcendence
degree, the prime spectrum of an algebra, localization, and the classical
theory of Noetherian and Artinian rings. Later chapters include some algebraic
theory of elliptic curves (featuring the Mordell-Weil theorem) and valuation
theory, including local fields.

One feature of the book is an extension of the text through a series of
appendices. This permits the inclusion of more advanced material, such as
transcendental field extensions, the discriminant and resultant, the theory of
Dedekind domains, and basic theorems of rings of algebraic integers. An
extended appendix on derivations includes the Jacobian conjecture and
Makar-Limanov's theory of locally nilpotent derivations. Gröbner bases can
be found in another appendix.

Exercises provide a further extension of the text. The book can be used both
as a textbook and as a reference source.

#### Reviews & Endorsements

The book is consistently organized in several layers (main text, supplements, appendices), which makes it a valuable source for readers of various levels, from graduate students to researchers.

-- European Mathematical Society Newsletter

Its outstanding features are of many kinds, ranging from its underlying philosophy of keeping the material as much to the point as possible, thereby being highly efficient, up to the vast amount of topical extras that are barely found somewhere else. Already this first volume must be seen as a didactic masterpiece, profitable for both students and teachers likewise.

-- Zentralblatt MATH

The author chose a very nice way to present all the basics needed to have a "solid basis in algebra", walking "direct to the goal" (my words), and showing the most important results of commutative algebra as soon and as simply as possible.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Graduate Algebra: Commutative View

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Introduction xi12 free
- List of symbols xv16 free
- Chapter 0. Introduction and Prerequisites 120 free
- Part I. Modules 3352
- Chapter 1. Introduction to Modules and their Structure Theory 3554
- Chapter 2. Finitely Generated Modules 5170
- Cyclic modules 5170
- Generating sets 5271
- Direct sums of two modules 5372
- The direct sum of any set of modules 5473
- Bases and free modules 5675
- Matrices over commutative rings 5877
- Torsion 6180
- The structure of finitely generated modules over a PID 6281
- The theory of a single linear transformation 7190
- Application to Abelian groups 7796
- Appendix 2A: Arithmetic Lattices 7796

- Chapter 3. Simple Modules and Composition Series 81100

- Part II. Affine Algebras and Noetherian Rings Introduction to Part II 99118
- Chapter 6. Transcendence Degree and the Krull Dimension of a Ring 171190
- Abstract dependence 172191
- Noether normalization 178197
- Digression: Cancellation 180199
- Maximal ideals of polynomial rings 180199
- Prime ideals and Krull dimension 181200
- Lifting prime ideals to related rings 184203
- Main Theorem B 188207
- Supplement: Integral closure and normal domains 189208
- Appendix 6A: The automorphisms of F[ Ai, 194213
- Appendix 6B: Derivations of algebras 197216

- Chapter 7. Modules and Rings Satisfying Chain Conditions 207226
- Chapter 8. Localization and the Prime Spectrum 225244
- Chapter 9. The Krull Dimension Theory of Commutative Noetherian Rings 237256
- Prime ideals of Artinian and Noetherian rings 238257
- Exercises – Part II 247266
- The Principal Ideal Theorem and its generalization 240259
- Supplement: Catenarity of affine algebras 242261
- Reduced rings and radical ideals 243262
- Chapter 4 247266
- Appendix 4A 257276
- Appendix 4C 262281
- Appendix 4B 258277
- Chapter 5 264283
- Chapter 6 264283
- Appendix 6B 268287
- Chapter 7 274293
- Appendix 7A 276295
- Chapter 8 277296
- Chapter 9 280299

- Chapter 4. Galois Theory of Fields 101120
- Field extensions 102121
- Adjoining roots of a polynomial 108127
- Separable polynomials and separable elements 114133
- The Galois group 117136
- Galois extensions 119138
- Application: Finite fields 126145
- The Galois closure and intermediate subfields 129148
- Chains of subfields 130149
- Application: Algebraically closed fields and the algebraic closure 133152
- Constructibility of numbers 135154
- Solvability of polynomials by radicals 136155
- Supplement: Trace and norm 141160
- Appendix 4A: Generic Methods in Field Theory: Transcendental Extensions 146165
- Transcendental field extensions 146165
- Appendix 4B: Computational Methods 150169
- The resultant of two polynomials 151170
- Appendix 4C: Formally Real Fields 155174

- Chapter 5. Algebras and Affine Fields 157176

- Part III. Applications to Geometry and Number Theory Introduction to Part III 287306
- Chapter 10. The Algebraic Foundations of Geometry 289308
- Chapter 11. Applications to Algebraic Geometry over the Rationals- Diophantine Equations and Elliptic Curves 313332
- Chapter 12. Absolute Values and Valuation Rings 339358
- Absolute values 340359
- Valuations 346365
- Completions 351370
- Extensions of absolute values 356375
- Supplement: Valuation rings and the integral closure 361380
- The ramification index and residue field 363382
- Local fields 369388
- Appendix 12A: Dedekind Domains and Class Field Theory 371390
- The ring- theoretic structure of Dedekind domains 371390
- The class group and class number 378397
- Exercises – Part III 387406

- List of major results 413432
- Bibliography 427446
- Index 431450
- Back Cover Back Cover1458