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Graduat e Algebra : Commutativ e Vie w This page intentionally left blank Graduat e Algebra : Commutativ e View

Louis Halle Rowen

Graduate Studies in

Volum e 73

KHSS^ K l|y|^| America n Mathematica l Societ y iSyiiU ^ Providence , Rhod e Islan d

Contents

Introduction xi

List of symbols xv

Chapter 0. Introduction and Prerequisites 1 Groups 2 Rings 6 9 Structure theories 12 Vector spaces and 13 Bilinear forms and inner products 15 Appendix 0A: Quadratic Forms 18 Appendix OB: Ordered 23

Exercises - Chapter 0 25 Appendix 0A 28 Appendix OB 31

Part I. Modules

Chapter 1. Introduction to Modules and their Structure Theory 35 Maps of modules 38 The of submodules of a 42 Appendix 1A: Categories 44 VI Contents

Chapter 2. Finitely Generated Modules 51 Cyclic modules 51 Generating sets 52 Direct sums of two modules 53 The of any set of modules 54 Bases and free modules 56 Matrices over commutative rings 58 Torsion 61 The structure of finitely generated modules over a PID 62 The theory of a single linear transformation 71 Application to Abelian groups 77 Appendix 2A: Arithmetic Lattices 77

Chapter 3. Simple Modules and Composition 81 Simple modules 81 82 A -theoretic version of composition series 87

Exercises — Part I 89 Chapter 1 89 Appendix 1A 90 Chapter 2 94 Chapter 3 96

Part II. AfRne Algebras and Noetherian Rings Introduction to Part II 99

Chapter 4. Galois Theory of Fields 101 extensions 102 Adjoining roots of a 108 Separable polynomials and separable elements 114 The 117 Galois extensions 119 Application: Finite fields 126 The Galois closure and intermediate subfields 129 Chains of subfields 130 Contents vn

Application: Algebraically closed fields and the 133 Constructibility of numbers 135 Solvability of polynomials by radicals 136 Supplement: Trace and norm 141 Appendix 4A: Generic Methods in Field Theory: Transcendental Extensions 146 Transcendental field extensions 146 Appendix 4B: Computational Methods 150 The of two polynomials 151 Appendix 4C: Formally Real Fields 155 Chapter 5. Algebras and Affine Fields 157 Affine algebras 161 The structure of affine fields - Main Theorem A 161 Integral extensions 165

Chapter 6. Transcendence Degree and the of a 171 Abstract dependence 172 Noether normalization 178 Digression: Cancellation 180 Maximal ideals of polynomial rings 180 Prime ideals and Krull dimension 181 Lifting prime ideals to related rings 184 Main Theorem B 188 Supplement: Integral closure and normal domains 189

Appendix 6A: The automorphisms of F[Ai,..., An] 194 Appendix 6B: Derivations of algebras 197

Chapter 7. Modules and Rings Satisfying Chain Conditions 207 Noetherian and Artinian modules (ACC and DCC) 207 Noetherian rings and Artinian rings 210 Supplement: Automorphisms, invariants, and Hilbert's fourteenth problem 214 Supplement: and filtered algebras 217 Appendix 7A: Grobner bases 220 Vlll Contents

Chapter 8. Localization and the Prime Spectrum 225 Localization 225 Localizing the prime spectrum 230 Localization to local rings 232 Localization to semilocal rings 235

Chapter 9. The Krull Dimension Theory of Commutative Noetherian Rings 237 Prime ideals of Artinian and Noetherian rings 238 The Principal Theorem and its generalization 240 Supplement: Catenarity of affine algebras 242 Reduced rings and radical ideals 243

Exercises - Part II 247 Chapter 4 247 Appendix 4A 257 Appendix 4B 258 Appendix 4C 262 Chapter 5 264 Chapter 6 264 Appendix 6B 268 Chapter 7 274 Appendix 7A 276 Chapter 8 277 Chapter 9 280

Part III. Applications to Geometry and Introduction to Part III 287

Chapter 10. The Algebraic Foundations of Geometry 289 Affine algebraic sets 290 Hilbert's Nullstellensatz 293 Affine varieties 294 Affine "schemes" 298 Projective varieties and graded algebras 303 Varieties and their coordinate algebras 308 Appendix 10A. Singular points and tangents 309 Contents IX

Chapter 11. Applications to over the Rationals- Diophantine Equations and Elliptic Curves 313 Curves 315 Cubic curves 318 Elliptic curves 322 Reduction modulo p 337

Chapter 12. Absolute Values and Valuation Rings 339 Absolute values 340 Valuations 346 Completions 351 Extensions of absolute values 356 Supplement: Valuation rings and the integral closure 361 The ramification index and residue field 363 Local fields 369 Appendix 12A: Dedekind Domains and 371 The ring-theoretic structure of Dedekind domains 371 The class group and class number 378

Exercises - Part III 387 Chapter 10 387 Appendix 10A 390 Chapter 11 391 Chapter 12 397 Appendix 12 A 404

List of major results 413

Bibliography 427

Index 431 This page intentionally left blank Introduction

This book and its companion, [Row2] (henceforth, referred to as "Volume 2"), grew out of a year-long graduate course in algebra, whose goal was to provide algebraic tools to students from all walks of mathe• matics. These particular students had already seen modules and Galois theory, so in the first semester we focused on affine algebras and their role in algebraic geometry (Chapters 5 through 10 of this book), and the second semester treated those aspects of noncommutative algebra pertaining to the representation theory of finite groups (Chapters 19 and 20 of Volume 2). The underlying philosophy motivating the lectures was to keep the material as much to the point as possible. As the course material developed over many years, I started putting it into book form, but also rounding it out with some major developments in algebra in the twentieth century. There already are several excellent works in the literature, including the books by N. Jacobson, S. Lang, and P.M. Cohn, but developments of the last thirty years warrant another look at the subject. For example, Zelmanov's solution of the restricted Burnside problem, affine Lie algebras, and the blossoming of quantum all took place after these books were written, not to mention Wiles' proof of Fermat's Last Theorem. During this time, research in algebra has been so active that there is no hope of encompassing all of the advances in one or two volumes, as evidenced by Hazewinkel's ongoing project, Handbook of Algebra. Nevertheless, several major themes recur, including the interplay of algebra and geometry, and the study of algebraic structures through their representations into matrices, thereby bringing techniques of linear algebra into play. The goal in these two volumes is to bring some harmony to these themes.

XI Xll Introduction

Since a thorough account of the interesting advances in algebra over the last 50 years would require a work of epic proportions, selections were made at every step. The task was further complicated when some respected col• leagues and friends pointed out that any text suitable for graduate students in the United States must contain a treatment of modules over a principal as well as the basics of Galois theory, which, together with their exercises, account for some 100 pages. So the project grew to two volumes. Nevertheless, the two volumes were planned and written together, and many results were included in the first volume for use in the second. Accordingly, a unified introduction seems justified. The first volume features (commutative) affine algebras, and aims quickly for the fundamental theorems from needed for affine algebraic geometry. This means a thorough treatment of transcendence de• gree and its relation to the lengths of chains of prime ideals in an affine algebra, and some theory of Noetherian rings. In the process, we take the time to develop enough module theory to present basic tools from linear algebra (e.g., rational form, Jordan form, Jordan decomposition) and also enough field theory for the applications later on in the two volumes. For example, Chapter 4 contains a thorough account of the norm and trace in a ; its appendices include transcendental field extensions, Luroth's theorem, and the discriminant and resultant. Other appendices along the way include Makar-Limanov's application of derivations to basic questions of affine geometry, and Grobner bases (related to computational algebra). The later chapters involve applications used in arithmetic and al• gebraic curves and surfaces, culminating in the algebraic aspects of elliptic curves and the underlying number theory. The second volume focuses on structures arising from matrices. The twentieth century saw the sprouting of algebraic structures — groups and Lie algebras are used throughout mathematics, and are unified in Hopf algebras — and these are treated, mostly in the context of representation theory. On the other hand, some of the most beautiful results in algebra in the last 50 years have involved interrelations among different structures, such as Gabriel's classification of indecomposable modules of finite representation type in terms of Dynkin diagrams from graph theory and , and Zelmanov's solution of the restricted Burnside problem from group theory by means of Lie algebras and Jordan algebras. The reader is offered a sample of these various theories as well. To keep within the allotted space and to permit the book to serve its original purpose as a general text, the following decisions were made: 1. The organization is at four levels — main text, supplements, appen• dices, and exercises. An effort is made to keep this hierarchy consistent, Introduction xin in the sense that theorems from the main text do not rely on results from the supplements or appendices; the appendices do not rely on results in the exercises, and so forth. As a result, almost every chapter branches out with at least one appendix related to its material. (The exception to this rule is Chapter 25 in [Row2], which requires the categorical framework supplied in Appendix 1A.) The exercises often are extensions of the text, containing material which I did not have the heart to exclude altogether, often with very extensive hints amounting to almost complete proofs. 2. Most of the theory is presented over algebraically closed fields of characteristic 0, although there is material on central simple algebras and Azumaya algebras (Chapters 24 and 25 in [Row2]), to indicate how algebra looks over more general commutative rings. 3. The material in the main text is handled in as elementary a fashion as possible. For example, categorical methods appear only in the appendices and exercises. 4. Far-reaching generalizations to noncommutative rings, although close to my heart, often are relegated to the exercises, if not cut altogether. 5. Homological methods, used throughout algebra in the last 50 years, are only touched on, and the student should supplement this book with a text on . Prerequisites are discussed in length in Chapter 0. The reader should be familiar with the basic structures of undergraduate algebra: groups and rings. Most of the requisite theory can be found in any standard undergrad• uate text, such as Herstein's Topics in Algebra or Gallian's Contemporary , referred to as [Gal]. When needed, results are cited from my undergraduate text (Algebra: Groups, Rings, and Fields), which is re• ferred to as [Rowl]. More details can be often be found in Jacobson's text Basic Algebra /, referred to as [Jal]. Some prerequisites, such as the Chinese Remainder Theorem, are needed throughout; others, such as Lagrange interpolation, are included as a conve• nient reference when needed. The hierarchy of the book also is followed here; quadratic forms, covered in Appendix A, are needed only in the appendices; likewise for ordered groups (Appendix B). As mentioned earlier, this volume developed from a one-semester course, that coverered the main parts (without appendices) of Chapters 1, 2 (through Theorem 2.31), 3, Chapter 5, Chapter 6 (through Lemma 6.39), Chapters 7 through 9, and Chapter 10 (as far as time permitted). One could easily add another semester with Chapter 4 and perhaps the remainder of Chapter 2. Parts of Chapters 2, 4, or 6 have been used at times in graduate student XIV Introduction seminars. I hope this book will prove useful to students who are looking for a solid basis in algebra. Many people have generously contributed of their time and energy to help bring this work to fruition. First thanks are due to Lance Small, who years ago suggested that this book be published with the AMS, and who pro• vided useful advice about topics. Most of an earlier draft of this manuscript was written during my Sabbatical visiting Darrell Haile at Indiana Univer• sity at Bloomington (in 2000(!)), and the final draft of Volume 1 was made while visiting David Saltman at the University of Texas at Austin, in 2005. I thank both Indiana University and the University of Texas for their gracious hospitality during these periods. Lenny Makar-Limanov graciously permit• ted free use of his unpublished notes on the use of derivations in studying affine geometry, and went over the material with me. Sue Montgomery ex• plained the newer results about the classification of finite dimensional Hopf algebras. Improvements and corrections were suggested by Jonathan Beck, Alexei Belov, Boris Kunyavski, Malka Schaps, Roy Ben-Ari, Eli Matzri, Tal Perri, and Shai Sarussi. But I would like to express special gratitude to David Saltman, editor of this AMS series, and Uzi Vishne, both of whom spent hours going through the text at various stages and suggested many crucial corrections and improvements. Saltman also explained some key points in the geometry involved. Prom the publishing side, Sergei Gelfand has been patient for years, while this work was maturing. Finally, as always, Miriam Beller has helped enormously with the technical preparation of the manuscript. List of Symbols

rning: Some notations have multiple meanings. A C B, N, N+, A^ 1

[G : H], Sn 3

sgn(7r), An, CG(A), Z(G) 4 GL(n, F) 5 x Mn(F), R , Cent(R), I

C[Ai,..., An] 10 F(X) 11 v* 13 dimF, e*, tr, det, V®W,V®W 14 (v,w), v -Lw, S1- 15

V = Wi ± W2, radF 16 ATI* 17

0(V, Q), Qi + Q2, (ai, •.., an), n(ai) 20

^~it; 21 AT

AnnR, M/K 40

A — B, CR(M) 42 a V b, a A b 43 Set 45

XV XVI Symbols

Mon, Grp, Ab, Ring, R-Mod, Mod-i?, C x V, A -4 B 46 coker/ 48 Fun(C,V) 50 Ra 51 isTei V 53 n^i 54

oMi, Mi e • • • e Aft, e/i 55 i?(") 57

Mn(i?), adj(^), | | 59 tor(M) 61

Afc(A), Afc(A) 69

Ts, Tn 75 C(M) 82 £(M) 85 #/F 99

Ia 103 /a, dega 105 ^A 109 char(F) 116 G&l(K/F) 117 Euler(n), Xs 120

ap 127

Fn 128

G", G«, 7i(G) 131

F, Fsep 134 Gal/ 137

trE/F, NF/F 142

F(Xl,...,\n),F(X) 146 disc/, disc B 150 A,(/) 151 #(/,

C[r],C[n,...,rn], C[5] 158 edep 173

: CD orq 05 J^ J fe # Co to Co > O n S3 Q Q > **] CO ^ s to i—• JS CD 53 w, <2 CD o ^ 'co t3 ti ^ C o ^> to ^ O 3 So a O n 5' •fl Orq CO c! 3 S^ S3 ta B. C o a- 'to CD > Orq

CO CD

cooococoootototot o to to to to to to to to to to to to to to to to to 1—* h^ CD ai Or Or hl^ OO GO h^ CO CO OOOOOCOCOCOC O ^ ^ to to to to1 o o o CO^ICOh^OCOO^tOh ^ O h-1 Oi Or O CO ^ to O CO -a Or I— CO ^ 00 to 00 -a XV111 Symbols

T(V)V 310

Cf, f 315 O, L(p,q), pq, p + q 319 E, E(F) 329 "(P), MP) 334 I I 340

vp 341 OF, PF, (i?) 349

Zp, Qp, F[[A]] 355 F((A)) 356 F 363 e = e(£/F), / = f(E/F) 364 JT(C) 373 C(C), CM(C) 378 CK, ||A|| 379 List of Major Results

The prefix E denotes that the referred result is an exercise, such as E0.5. Since the exercises do not necessarily follow a chapter immediately, their page numbers may be out of sequence.

Chapter 0.

Correspondence between subgroups of G and G/N. 2

Cauchy's theorem, Abelian case. 3

0.(y. The center of a ^-group is nontrivial. 4

(Sylow's First Theorem). Any finite group G has a subgroup of any prime power order dividing |G|. 5

0.1. (Noether's first theorem). Any ring (p: R —> T induces an injection R/kenp c—> T. 6

0.2. Correspondence between ideals of R and R/L 7

0.2'. C + I)/I^C/(Cnl). 7

0.3 (Chinese remainder theorem) For any comaximal ideals Ai,..., At of i?, the homomorphism R —> RjA\ x • • • x R/At is onto. 8 0A.5. In a nonsingular quadratic space, any totally isotropic subspace of dimension m is contained in a hyperbolic space of dimension 2m. 21

E0.5. Any additive submonoid of N is finitely generated. 25

E0.27. Any subgroup of finite index in a f.g. group is f.g. 28

413 414 Major Results

Appendix OA.

0A.10 (Witt cancellation theorem). Any quadratic space is an orthogo• nal sum of a totally isotropic space, a hyperbolic space, and an anisotropic space. 22

0A.11 (Witt cancellation theorem). Any common quadratic space can be cancelled from an isomorphism of orthogonal sums. 22

E0A.12 (Cartan-Dieudonne). In a nonsingular quadratic space of di• mension n, every isometry is a product of at most n reflections. 29

Chapter 1.

1.13. Noether's isomorphism theorems for modules. 41

1.17. The modules of M/K have the form N/K for K < N < M. 42

1.20 (Modularity property). (Ni + K) n N2 = Ni + (K n N2) when iVi C N2. 44

Chapter 2.

2.2. A module M is cyclic iff M ^ R/L. 52

2.16. Identification of direct sum with internal direct sum. 56 2.30. There is a 1:1 correspondence between maps R^ —+ R^ and n x m matrices. 60 2.31. i?(n) = i?(m) for R commutative implies m — n. (Improved in Exercises 2.7-2.9.) 60

2.40. Over a PID, any submodule of a free module is free. 62

2.52. Over a PID, any matrix is similar to a matrix in normal form. 66

2.54. Any f.g. module over a PID is a direct sum of cyclic submodules. (See also Theorem 2.61 and Exercise E2.17.) 68

2.60. Determination of the entries of the diagonal matrix of Theorem 2.52, in terms of the cofactors of the original matrix. 70

2.61. The direct sum decomposition in Theorem 2.54 is unique. 70

2.64. The rows of the Hamilton-Cayley matrix of a transformation comprise a base for its module structure. 71 Major Results 415

2.65 (Hamilton-Cayley, Frobenius). The minimal polynomial of a lin• ear transformation divides the characteristic polynomial, and has the same irreducible factors. 73

2.69 (Jordan canonical form). Any linear transformation (over a field containing its eigenvalues) is a direct sum of Jordan blocks. 74

2.72ff. (Jordan decomposition). T = Ts + Tn, where Ts is semisimple and Tn is nilpotent. 75f. 2.77 (Fundamental theorem of f.g. Abelian groups). Every finitely gen• erated is a finite sum of a f.g. free Abelian group and finite cyclic groups of prime power order. 77

E2.5. Equivalent conditions for a module M to be isomorphic to a direct sum K ®N. 94

E2.7, E2.9. If R is commutative and there is a map R^m) -> #(n) that is onto (resp. 1:1), then m>n (resp. m < n). 94f.

Chapter 3.

3.11 (Jordan-Holder-Schreier theorem). If M has a composition series, then any chain can be refined to an equivalent composition series, and com• position length is additive. (See also Exercise E3.2.) 85

E3.5 (Jordan-Holder-Schreier theorem for groups). If a group G has a composition series, then any subnormal chain can be refined to an equivalent composition series. 96

Chapter 4.

4.1. Any finite subgroup of a field is cyclic. 102

4.4. Dimensions of a chain of field extensions are multiplicative. 103

4.9. A polynomial / satisfied by a is irreducible, iff / is the minimal polynomial of a. 106

4.13 The composite of algebraic extensions is algebraic. 108

4.18 Automorphisms of a field permute the roots of a polynomial, and, conversely, if a2 G F[a{\ is a root of the minimal polynomial of ai, there is an automorphism sending a\ to a

4.23. Any polynomial of degree n has a splitting field of degree

4.26. Any isomorphism of fields extends to their splitting fields. 113 416 Major Results

4.31. A polynomial / is separable, iff / and /' are relatively prime. 115 4.44. The degree of a Galois field extension equals the order of the Galois group of the extension. 118 4.55. Equivalent conditions for a extension to be Galois. 122 4.56 (Artin's Lemma). [E : EG] < \G\ for any finite group G of auto• morphisms of E. 123 4.58 (Dedekind independence). Distinct to a field are linearly independent. 123 4.61 (Fundamental theorem of Galois theory). The intermediate fields L of a E/F are in 1:1 correspondence with the subgroups of Gal(F/F), in reverse order; L/F is Galois iff Gsl(E/L) < Gal(F/F). 124 4.64. The order of any finite field is a prime power. 127 4.66. For any prime power n, there is a unique field of order n (up to isomorphism), which is comprised of the set of roots of An — A. 127 4.70. Any finite field is cyclic Galois over its characteristic subfield. 128 4.79, 4.80. (Primitive root theorem) A finite field extension K/F is separable iff K has the form F[a], iff K/F has a finite number of intermediate fields. The number of F-embeddings of K into the Galois closure equals [K:F\. 130 4.87 (Fundamental theorem of algebra). C is algebraically closed. 133 4.88. Existence of the algebraic closure. (See also Exercise E4.37.) 134 4.95 (Hilbert's Theorem 90). In a cyclic field extension, N(c) — 1, iff c=-^. 138

4.96. Structure of Kummer extensions. 138 4.98-4.100. A polynomial / is solvable by radicals, iff its Galois group is solvable; this occurs whenever deg / < 4, but there is a counterexample for deg/-5. 139f. 4.108. Transitivity of norm and trace. 144 E4.12. e is transcendental over Q. 248 E4.13. 7T is transcendental over Q. 248 E4.22. The Galois group of the cyclotomic extension is Euler(n). 250 Major Results 417

E4.30. Formula for the number of irreducible polynomials of degree m over Z/p. 251 E4.37. The algebraic closure is unique up to isomorphism. 251 E4.40-E4.43. Impossibility of doubling the cube, circling the square, or trisecting an angle by means of classical constructibility. 252 E4.44. Khayyam's solution of the . 252 E4.50. The regular n-gon is constructible iff n is a product of a power of 2 with distinct Fermat primes. 252 E4.54. The Tartaglia-Cardano solution of the cubic equation. 253 E4.55. Ferrari's solution of the cubic equation. 253 E4.60. The polynomial / = A5 — pqX + p for p, q prime has Galois group S5 and thus is not Galois solvable. (See also Exercise E6.13 for an approach via reduction mod p.) 253 E4.65. A field extension is separable iff the trace map is onto, iff the trace bilinear form is nondegenerate. 254 E4.67 (Hilbert's Theorem 90, additive version). In a cyclic field exten• sion, tr(c) = 0 iff c = a — a (a). 254 E4.80. Any field extension is a purely inseparable extension of a sepa• rable extension. 255 E4.82. The separability index and inseparability index are multiplica• tive. 256 E4.83, E4.84. Any normal field extension is a of a purely inseparable extension (and thus can be decomposed as a product of a separable and purely inseparable extension). 256 E4.89. Gauss' quadratic reciprocity law, Swan's proof. (See Exer• cise E12A.17 for a proof using ramification theory.) 257 Appendix 4A. 4A.5 (Luroth). Any subfield of F(X) is purely transcendental. 148 E4A.1. The elementary symmetric functions generate the algebra of invariants (with respect to the symmetric group Sn). 257 E4A.2 (Noether's problem). The fixed algebra of a purely transcendental extension F(Ai,..., An) with respect to an automorphism acting cyclically 418 Major Results on the indeterminates is purely transcendental, provided F contains a prim• itive nth root of 1. (See Exercise E7.5 for a counterexample when the action is not cyclic.) 257

Appendix 4B.

4B.11. The resultant of two polynomials is 0 iff they are not relatively prime. 152

4B.12. A formula for the resultant of two polynomials. 153

E4B.6 (Bezout). Any two nonconstant homogeneous polynomials in at least three indeterminates over an algebraically closed field have a common nontrivial zero. (See also Exercise E10.17) 258f.

E4B.7 (Bezout). Over an algebraically closed field, any two nonconstant polynomials /,g having more than degfdegg points in their intersection have a common nontrivial factor. (See also Exercise E10.19) 259

E4B.13. The inverse of the Bezout matrix is constant on its reverse diagonals. 260

E4B.24. Any field of transcendence degree m over an algebraically closed field is a Cm-field. 262

Appendix 4C.

E4C.10. Any formally real field is contained in a real closed subfield K of F, such that F = K[^l. 263

E4BC.15. Any order-embedding from F to a real-closed field extends uniquely to an order-embedding from the real closure of F. 264

Chapter 5.

5.11 (Main Theorem A). Any affine field is finite dimensional. (A second proof is given on page 170. See Exercise E5.5 for a quick proof when the base field is uncountable.) 162

5.15 (Artin-Tate). Any subalgebra of finite index in an affine algebra is itself affine. (See also Exercise E7.4) 163

5.21. An element r is integral over C iff C[r] is f.g. over C. 166

5.27. A composite of integral extensions is integral. 168 Major Results 419

Chapter 6.

6.8. Any two bases have the same cardinality. 176

6.10 (Noether normalization). Any affine algebra is integral over a suit• able subalgebra that is isomorphic to a polynomial algebra. (See also Exer• cises E6.2, E6B.8, and E9.13.) 178

6.14. The maximal ideals of F[Ai,...,An] correspond to the points ofFK 181

6.28. INC holds for integral extensions. 186

6.32. LO and GU hold for integral extensions. (A second proof is given in Exercises E8.24 and E8.25) 187

6.35 (Main Theorem B). The Krull dimension equals the transcendence degree, for affine domains. 188

6.37. Every prime ideal of an affine algebra is the intersection of maximal ideals. 188

6.44 (Gauss' lemma). If the product fg of monic polynomials is in C[A], for C a UFD, then f,ge C[X\. 190

6.47. GD holds in any integral extension of a normal domain. 191

6.49. Every of an affine domain has height equal to the Krull dimension (and thus also equal to the transcendence degree). 192

6.50 (Noether). In characteristic 0, any integral extension of an affine domain in a finite field extension is a f.g. module and thus Noetherian. 192

6.51 (Noether). Any integral extension of a normal domain in a finite separable field extension is a f.g. module and thus Noetherian. 193

6.52. The integral closure of a PID C in a finite separable field extension is f.g. free as a C-module. 193

E6.2. Noether normalization over an arbitrary field. 265

E6.9. A prime ideal of height 2 that cannot be generated by two ele• ments. (Compare with Exercise E10.12.) 265

E6.16. Equivalent conditions for a finitely generated (not necessarily finite) field extension to be separable. 267 420 Major Results

Appendix 6B. 6B.5 (Leibniz' rule). Sn(ab) = J2 (;)^(a)^(&). 198 6B.10. Any derivation extends uniquely to a separable algebraic exten• sion. 199f. 6B.13. Any derivation extends to a purely transcendental extension. (See also Exercise 6B.5) 200f. 6B.16. R6 is algebraically closed in R. 201 6B.28. Nonisomorphic algebras of the same transcendence degree may have the same ring of absolute constants. (See also Exercises E6B.31- E6B.33.) 204 6B.29. If R is a C-domain of transcendence degree 1, and has an LND, theni?^C[A]. 205 E6B.15. All automorphisms of C[Ai, A2] are tame. 270

E6B.28. AK(i2[Ai,..., An]) = AK(i?), for any C-domain R of transcend• ence degree 1. 273 Chapter 7. 7.3. A module is Noetherian (resp. Artinian), iff its submodules satisfy the maximal (resp. minimal) condition. 208 7.7. A module is Noetherian, iff every submodule is f.g. 209 7.10. Every f.g. module over a left Noetherian (resp. Artinian) ring is Noetherian (resp. Artinian). 210 7.17. Affine algebras are Noetherian. 212 7.18 (Hilbert's basis theorem). The ring of polynomials over a Noether• ian ring is Noetherian. (See also Exercises E7.14 and E7A.2) 212 7.19. (Fundamental theorem of invariant theory) The fixed subalge- bra of an affine algebra under a finite group of automorphisms is affine. (See also Exercises E7.7 and E7.9.) 214 7A.3. If a graded ring has ideals N C A whose leading terms are the same, then N = A. 222 E7.1. An Artinian module that is not Noetherian. 274 E7.9. A counterexample to Hilbert's fourteenth problem. 275 Major Results 421

Chapter 8.

8.2. The construction of the localization S~lR is an algebra. (See also Exercises E8.2 and E8.4.) 226

8.8. There is a lattice injection from the ideals of S_1R to the ideals of R. 229

8.11. The prime spectrum of S_1R is homeomorphic to that part of Spec(i?) disjoint from S. 230

8.19. Equivalent conditions for a ring to be local. 232

8.24 ("Nakayama's lemma"). PM ^ M for every f.g. module M over a R having maximal ideal P. (Improved in Exercise E8.22.) 234

Chapter 9.

9.2. The product of prime ideals in a is 0. (Generalized in Exercise E9.1) 238

9.4. A Noetherian ring has Krull dimension 0 iff it is Artinian. 239

9.7. ( theorem = PIT). Every prime ideal minimal over a principal ideal has height < 1. 240

9.9 (Generalized PIT). Every prime ideal minimal over an ideal gener• ated by t elements has height < t. 242

9.10. Every prime ideal of a Noetherian ring has finite height. 242

9.11 (Catenarity). All saturated chains of prime ideals of an affine do• main have the same length. 243

9.21. An ideal of a Noetherian ring is radical, iff it is the intersection of finitely many prime ideals. (Generalized in Exercises E9.24 and E9.25.) 245

E9.4 (Eakin-Nagata). If a Noetherian ring is f.g. over a (central) C, then C is also Noetherian. 280

E9.7 (Krull intersection theorem). DAnM = 0 for any f.g. module M over a Noetherian ring R with A < R. 281

n E9.9. nne^A — 0, for any ideal A of a Noetherian domain. 281

E9.10 (Nagata). An example of a Noetherian ring having infinite Krull dimension, although all prime ideals have finite height. 281 422 Major Results

E9.10. In a Noetherian ring, every height 2 prime ideal contains infinitely many height 1 prime ideals. 281

E9.16. Every regular local ring is an . 282

Chapter 10.

10.11 (Hilbert's Nullstellensatz). Every radical ideal of F[Ai,...,An] comes from an algebraic set. (See also Exercise El0.1.) 294

10.14. The affine subvarieties of A^n) correspond to the prime ideals of F[Aj,...,An]. 295

10.16. Every (affine) algebraic set is a finite union of varieties. 295

10.20. The dimension of an affine variety is the Krull dimension of its coordinate algebra. 296

10.33. The local ring of a point of an affine variety V is the localization of the coordinate algebra at the prime ideal corresponding to V. 301

10.36. For any morphism : [/ —> W of affine varieties, and any open subset U of F, $(U) contains an open subset of the closure of &{V). 302

El0.7. An affine curve is a line iff its coordinate algebra has AK equal to C. 388

E10.8 (Abhyankar-Eaken-Heinzer). Cancellation for affine curves. 388

E10.17 (Bezout). Any two planar projective curves have a common point. 389

E10.18 (Bezout). If two planar projective curves for /, g have more than ran points in their intersection where ra, n are the respective degrees, then / and g have a common factor. 389

Appendix 10A.

10A.7. The tangent space T(V)W is identified with Deriv(Ov, F). 311

E10A.4. Computation of the rank of the Jacobian matrix. 391

E10A.6. A variety is nonsingular at a point v, iff its local ring Ow is regular. 391

Chapter 11.

11.4. Any conic with an F-point is parametrizable. 317 Major Results 423

11.6. Any cubic with an F-singular point is parametrizable. 318

11.9. Any nonsingular cubic curve has an Abelian group structure. (A different, analytic, proof is outlined in Exercise Ell.30.) 319

11.16. Any elliptic curve can be put in the form / = y2 — q(x), where degq = 3. 324

11.21. A criterion for when p = 2q in an elliptic curve. 327

11.22 (Mordell-Weil). The group of an elliptic curve defined over an is finitely generated. 330

El 1.13. Every nonsingular planar curve of degree > 2 has a flex. 392

El 1.21 (Fermat). A rational right triangle cannot have area 2. 393

El 1.24. The Weierstrass function is an even elliptic function with a double pole at 0. 394

El 1.40, El 1.42 (Lurz-Nagell). Properties of the torsion elements of the group of an elliptic curve. 397

Chapter 12.

12.6. Conditions for absolute values to be equivalent. 343

12.10. An absolute value is nonarchimedean iff it is bounded on the characteristic subfield. 345

12.25. The ideals of a are ordered under inclusion. 351

12.37. If a field F is complete with respect to an absolute value, then the extension to any finite field extension is unique. 358

12.43. Any nonarchimedean absolute value can be extended to a finite field extension. 361

12.47. The localization of a valuation ring at a prime ideal of its integral closure (in a finite field extension) is also a valuation ring. 362

12.54. ef<[E:F]. 364

12.57. The valuation ring with respect to a discrete valuation is a PID. 366

12.58. If F is complete with respect to a discrete valuation v, then v extends uniquely to any finite field extension E, and ef = [E : F]. 366 424 Major Results

12.62 (Hensel's lemma). If F is Henselian then monic factorizations lift up from polynomials over its residue field. 368 12.65. Any finite field extension of a F is totally ramified of dimension e over the unique maximal unramified intermediate field, which is cyclic Galois of dimension / over F and is generated by a root of 1. 369 E12.6 (Artin-Whaples). For any finite set of t inequivalent nontrivial absolute values on a field F and any elements ai,..., at G F, there is a G F with \a — di\i arbitrarily small, 1 < i < t. 397 E12.7. Every archimedean absolute value on Q is equivalent to the ordinary absolute value. 398 E12.9. R and C are the only complete fields with respect to an archi• medean absolute value. 398 E12.28 (Kurschak). The completion of an algebraically closed field is algebraically closed. 400 E12.30. Newton's approximation method for locating roots of a polyno• mial. 400 El2.36. Any absolute value on a complete field extends uniquely to a finite Galois extension. 402 Appendix 12A. 12A.5. In a , every prime ideal is maximal, every is invert ible, and the factorization of an ideal as a product of prime ideals is unique. 373 12A. 13. An integral domain is Dedekind iff it is Noetherian and normal of height 1. 375 12A. 14. A Dedekind domain having only finitely many prime ideals is a PID. 376 12A.18. Computation of the dimension of the fields of fractions of an extension of Dedekind domains, in terms of the ramification indices. 378 12A.27 (Class number theorem). The class number of any ring of alge• braic is finite. (The class number is estimated in Exercises E12A.18, improved using E12A.22, Minkowski's constant.) 381 12A.31. If C is a Dedekind domain with finite class number and finitely generated group of units, then one can localize at a suitable element to obtain a PID with finitely generated group of units. 383 Major Results 425

12A.33 (Dirichlet unit theorem). The unit group of the of any algebraic number field is finitely generated. (This is made more precise in Exercise E12A.27.) 384

E12A.4. A Dedekind domain is a UFD iff it is a PID, iff every prime ideal is principal. 404

E12A.10. A prime ideal pZ of Z ramifies in a Dedekind domain R iff p divides the discriminant. 405

E12A.31. Any quadratic form of dimension >5 over Qp is isotropic. 409 E12A.48 (Hasse-Minkowski). A quadratic form with rational coefficients is isotropic over Q, iff it is isotropic both over R and over Qp for every prime p. 41 If.

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absolute value, 340ff. algebraic set, 290 archimedean, 344, 398f. affine, 290ff. equivalent, 342, 357 ideal of, 292ff. extension of, 356ff., 401f. algebraically closed, 133, 261, 400 nonarchimedean, see nonarchimedean quasi-, 261 on Q, 346, 398 algebraically dependent, 172, 174 ordinary, 341 algebraically independent, 172, 269 p-adic, 341f., 399 almost all, 1 normalized, 342 alternating group, 4 trivial, 341 ambiguity, 223 abstract nonsense, 228 anisotropic, 17 ACC, 207 annihilator, 40 adjoint, 17 Artin's lemma, 123, 249 affine Artin-Rees lemma, 280 algebra, 161ff., 212, 221ff., 264 Artin-Tate lemma, 163, 264, 274 domain, 161, 171, 188, 243 Artin-Whaples approximation, 397 Artinian geometry, 298f. scheme, 298f. module, 208ff., 274 variety, see variety, affine ring, 210f., 238, 239 affinization, 307 ascending chain condition, see ACC algebra, 99, 144, 157ff. automorphism, 6, 110, 117ff., 138, 194, automorphism, 159 269ff. filtered, 219, 276 Nagata's, 195, 271 scalar, 194 graded, see graded algebra tame, 194, 270 homomorphism, 159 stably, 195 isomorphism, 159 triangular, 194, 270 represent able, 142 wild, 195 algebraic, 104, 105, 162, 165, 177 axiom of choice, 2 closure, 134, 251f. separable, 134, 255 algebraic group base, 13, 176, 264 linear, 291 of free module, 56 reductive, 216 of vector space, 14 linearly, 216 orthogonal, 16 algebraic integers, 339, 379 standard, 65 algebraic number field, 330, 379ff. transcendence, see transcendence base 432 Index

Bezout matrix, 259f., 389 conjugacy class, 4 Bezout's theorems, 258ff., 261 conjugate, 4 bilinear form, 15ff., 19 constructible, 135, 252 alternate, 15, 17 convex, 79 diagonalizable, 16 coordinate algebra, 295f., 298f., 308f., 388 Hermitian, 16 coproduct, 91 nondegenerate, 15 countable, 247 nonsingular, 15 CRT, see Chinese Remainder Theorem positive definite, 16 cubic symmetric, 15 curve, 318ff., 393 birational equivalence, 308, 389 twisted, 388 Buchberger's algorithm, 224 nonsingular, see nonsingular cubic curve cancellation question, 180, 267f., 388 equation, 252, 253, 258 cardinal, 1 curve, 305, 313f., 315ff. category, 44ff. conic, 317f. Abelian, 91 cubic, 318 dual, 48 defined over F, 314 pre-additive, 47 elliptic, see elliptic curve small, 45 planar, 315 catenarity, 242f. rational, 315 Cauchy sequence, 351 cycle, 3 center cyclotomic of group, 4 extension, 107, 112, 120, 249f., 365 of ring, 6 polynomial, 249 centralizer, 4 chain, 2 DCC, 207 ascending, 2 on prime ideals, 242 descending, 2 decomposition group, 265f., 402 equivalent, 84 Dedekind length of, 82, 182 domain, 37Iff. of subgroups, 87 independence theorem, 123, 249 subnormal, 87 ring, 371 of length t, 82 degree of modules, 82 of algebraic element, 103 of prime ideals, 182ff., 211 of field extension, 103 saturated, 242 of monomial, 18 characteristic, 116 of rational curve, 315 subfield, 126 with respect to LND, 202 Chinese remainder theorem, 8 dependence relation, (strong) abstract, 173 class dependent, 173 group, 378ff. algebraically, see algebraically dependent number, 378, 381ff., 406ff. derivation, 197ff., 268ff., 309ff. coefficient, 9 equivalent, 269 cofactor, 70 Jacobian, see Jacobian derivation cokernel, 54, 90 locally nilpotent, see LND commutative Nagata's, 271 diagram, 42, 89 on field extension, 199, 268f. square, 42 on , 200 subgroup, 131 triangular, 271 completion, 352fL, 358, 399f. descending chain condition, see DCC composition , 14, 290 length, 85 dimension, 14 series, 82ff., 209 Krull, 182, 187, 238ff., 265, 281, 375 congruence, 26 of an affine variety, 296 conic, 317f., 391f. direct section, 289 limit, 92, 279 Index 433

product, 54, 159 purely inseparable, 255, see also sum, 53, 55 inseparable internal, 53 separable, see separable extension Dirichlet unit theorem, 384 transcendental, see transcendental discriminant extension of base, 151, 266, 383 F-, 99, 109 of bilinear form, 17 finite, 126ff., 251, 390 of elliptic curve, 325, 393, 396 Henselian, 367f. of polynomial, 150, 153f. intermediate, 124ff., 249 of ring of algebraic integers, 383, 405, 407 local, 369ff., 403 domain, F-, 99, 108, 177 of fractions, 162, 200, 225, 340 doubly periodic, 328 of coordinate algebra dual of rational functions, 146, 299 base, 14 perfect, 117 category, 48 real-closed, 263ff. lattice, 43 residue, 363 numbers, 247 filter, 26 poset, 2 cofinite, 27 DVR, see valuation ring, discrete filtered product, 27 , 219 exhaustive, 219 e, 248 positive, 219 Eakin-Nagata-Formanek theorem, 280 finitely generated, 93, see also module, eigenspace, generalized, 75 finitely generated eigenvalue, 74f. flex, 322f., 392 elliptic form, 18 curve, 322ff., 393ff. quadratic, see quadratic form over C, 328, 393ff. normic, 261 Weierstrass form, 323f. formally real, 155 function, 328 free, 13, 26, 279 embedding, 103, 109 Abelian group, 77 epic, 46, 89 module (of rank n), 57f., 62, 94, 95 split, 53 Fitting decomposition, 73 , 9 Frobenius 73, Euler number, 250 automorphism, see Frobenius map exact sequence, 41, 53, 89 map, 127 short, 41, 53 morphism, 390 split, 53 of rational functions, 11 exclusion principle, 8, 25 problem, 25 prime, see prime exclusion functor, 49 forgetful, 49 factor (of chain), 82, 87 identity, 49 faithful, 159, 280 fundamental domain, 78 f.g., see module, finitely generated fundamental parallelogram, 328 Fermat prime, 136, 250f. Fundamental Theorem Fermat's Last Theorem, 323, 393, 405 of Abelian groups, 77 field, 6, 10ff., 99ff., 162, 169, 314, 339ff. of algebra, 133 algebraic closure of, see algebraic closure of Galois theory, 124 complete, 352, 358, 366, 400 of invariant theory, 214 Cn-, 261f., 404 extension, 99, 102ff. 117ff, 162, 318, 333, Galois, 101 356ff. closure, 129ff. algebraic, 105, 178, 247 correspondence, 102, 124ff., 249 cyclic, 254, 369 extension, 119ff., 142, 146 finite, 103, 247, 358, 361, 366 generic, 147 inseparable, see inseparable extension group, 117ff., 124fL, 137fi\, 193, 266 normal, 121 solvable, 132 434 Index

Gauss' lemma, 107, 190, 202 F-, 103 gcd, 9 substitution, 104, 160, 180 generated, 13, 52 hyperbolic plane, 20 generic method, 11, 107, 153 hyperbolic space, 20 going down GD, 191 hyperplane, projective, 307 going up, see GU hypersurface, 291 gr-prime, 218 gr-maximal, 305 ideal, 6 graded class, fractional, 378 algebra, 217, 276, 304ff. fractional, 350, 372ff., 379, 404 associated, 217, 276 invertible, 372ff., 404 domain, 218, 305 principal, 350, 405 homomorphism, 217 left, 6 ideal, 218, 221ff., 304 maximal, see maximal ideal module, 218 prime, see prime ideal associated, 219 principal, 189 submodule, 218 proper, 6 Grobner basis, 220ff. radical, 244, 283, 294ff. group, 2ff., 13 right, 6 Abelian, 13, 36, 77, 88, 319 incomparability (INC) 184ff. action, 5, 193, 279 independent, 173, 176 linear, 214 algebraically, see algebraically permutation, 214 independent alternating, see alternating group infinum, 43 dihedral, 13 injection, 6 Galois, see Galois group inseparable extension, 117, 145, 200, 254 general linear, 5 integral domain, 7, 69ff., 82, 162, 169, 183, nilpotent, 131 200f., 211, 226, 233, 340, 347, 371ff. of an elliptic curve, 319ff., 395 normal, 190ff., 266, 361f., 375f. of units, 250, 333, 349, 383, 408 integral orthogonal, see orthogonal group closure, 189, 193, 231, 361ff., 377, 379 p-, 4, 132 element, 165 profinite, 135 extension, 165fL, 178, 186ff., 231 solvable, 131, 139 over, 165 symmetric, see symmetric group integrally closed, 190, 231 GU, 185ff. invariant, 214, 257 inverse Hamilton-Cayley theorem, 25, 73 limit, 92, 279 Hasse principle, 338 system, 92 weak, 412 irreducible Hasse-Minkowski theorem, 411 algebraic set, 294 Hensel's lemma, 367f., 400 element, 9, 189 Hermite's theorem, 260 polynomial, see polynomial Hessian matrix, 392 topological space, 379f. Hilbert isometry, 18, 19 basis theorem, 212, 222, 276 isomorphism, 6, 46 fourteenth problem, 215f., 275f. F-, 103 generalized, 215f. of categories, 49 Nullstellensatz, 293ff., 387, 390 isotropic, 17 reciprocity law, 410 totally, 21 symbol, 409 Theorem 90, 138 Jacobian, 195 additive version, 254 derivation, 202, 269 homogeneous, 217 matrix, 195, 269, 391 polynomial, see polynomial Jordan homogenization, 307 canonical form, 74 homomorphism, 6 decomposition, 75, 96 Index 435

multiplicative, 76 similar, 64 Jordan-Holder-Schreier theorem, 85, 96 matrix, 63, 71 for groups, 96 diagonal, 66, 75, 291 nilpotent, 74 Kantor's diagonalization trick, 247 orthogonal, 291 kernel, 6, 12, 39, 40 special, 291 categorical, 48, 90 permutation, 65 Khayyam, 252 rank, 71 Krasner's lemma, 400 relations, 63 Kronecker delta, 1 symplectic, 291 Krull intersection theorem, 281 transposition, 65 Kummer extension, 137 unipotent, 76 matrix unit, 59 Lagrange maximal ideal, 7, 8, 9, 109, 164f., 180f., interpolation, 10 183, 187, 188, 189, 192, 234, 238, 265, resolvent, 254 266, 277, 282, 298, 359f., 362f., 374ff. lattice (poset), 43 height of, 183, 192 directed, 92 maximal principle, 2 homomorphism, 43 maximum condition, 208 of ideals, 229f. minimum condition, 23f., 31, 208 of submodules, 42ff. Minkowski's constant, 407 lattice (arithmetic), 78, 383, 407 modularity property, 44, 89 period, 329 module, 35fL, 59ff. Laurent series, 355, 399 Artinian, see Artinian module leading term, 221 cyclic, 51, 68, 81 Legendre symbol, 156 factor, 40 Leibniz' rule, 198 filtered, 219 Lie product, 198 finitely generated, 52fL, 62ff., 209, 234 lies above (local ring), 359 free, see free module line at infinity, 315 graded, see graded module linear transformation, 7Iff. homomorphism, 38 characteristic polynomial of, 73 isomorphism, 39 minimal polynomial of, 73 Noetherian, see linearization, 16 right, 35, 49 linearly disjoint, 266 simple, 8Iff., LND, 200ff., 270ff. monic LO, 184ff., 278 map, 39 local-global principles, 370f., 410ff. morphism, 46, 90 local ring, 232ff., 282, 350, 359 monoid, 13 of a point, 300 monomial, 18, 221 regular, 282, 391 Mordell-Weil theorem, 330ff., 396 localization inductive, 330 of fields, 369 weak, 330f. of modules, 279 morphism, 44ff., 90 of rings, 225ff., 277, 362, 377 compatible, 45 Luroth's theorem, 147f., 257 of functors, 50 Lutz-Nagell theorem, 397 of tangent spaces, 311 lying over, see LO of varieties, 301ff., 389 unit, 45 Main Theorem A, 162ff., 170, 264 zero, 47 Main Theorem B, 188 Makar-Limanov's theory of LND, 201ff., Nakayama's lemma, 234, 278, 281 271ff. natural isomorphism, 50 map, 38 natural transformation, 50 right multiplication, 39 Neron height function, 333ff. matrices, 58, 290f. Newton's method, 400 equivalent, 64 Newton's polytope, 401 436 Index

Noether isomorphism theorems, 6 even, 3 for algebras, 160 odd, 3 for modules, 41 pi, 248f. for monoids, 89 PID, 9, 52, 62fL, 82f, 95, 193, 239, 366, 376, Noether I, 6 379, 382, 404 Noether normalization, 168, 178, 267, 268f., PIT, see 281 point, Noether's problem, 257 F-, 314, 317 Noetherian finite, 315 domain, 281f., 375f. inflection, see flex module, 208ff., 274, 282 nonsingular, 309 ring, 210ff., 229, 238ff., 280ff., 375f. of A

projective scalar multiplication, 35 algebraic set, 304 , 25 irreducible, 304 semisimple, 75 n-space, 303 separability index, 256 transformation, 303 separable variety, see variety, projective element, 117 pullback, 93 extension, 117, 129f., 145, 193, 200, 268f. pushout, 94 of arbitrary dimension, 266f. Pythagorean triples, 317, 391 polynomial, see polynomial, separable sequence, quadratic exact, see exact sequence extension, 106, 136 M-, 282 form, 18ff., 28ff., 392 signature, 12 over local field, 408ff. singular (at a point), see point formula, 102 solvable map, 19, 21 by radicals, 102, 136ff. reciprocity law, 256f., 406, 409, 410 spans, 52 space, 19ff. spectrum, see prime spectrum geometric, 293 quadratically defined, 135, 252 splitting field, lllff., 119ff., 127, 133f., 137, quartic equation, 253 326 quasicompact, 278, 296 Steinitz exchange axiom, 173 structure, 12, 26 radical (of ideal), 244, 283 constants, 164 radical (of vector space), 16 subalgebra, 157 ramification index, 364, 378, 403 generated by, 158 ramification of prime ideals, 405f. subcategory, 45 ramified, totally, 364, 369, 405 characteristic, 126 rational form, 73 fixed, 120ff. rational function, 299 full, 45 domain of definition, 300, 306 subgroup, 2,3, 78, 102 grade-0, 306 cyclic, 102 regular on an open set, 300, 306 of finite index, 3, 28, 78 real closure, 263ff. p-Sylow, 4 reciprocity law, 370, 406, 410 sublattice, 78 reduction modulo p, 250, 266, 337 submodule, 37ff., 62 refining (a chain), 84 cyclic, 51 regular element, 95 generated by, 38 regular local ring, see local ring submonoid (multiplicative), 186, 225ff. regular n-gon, 135f. substructure, 12 regular representation, 144 subvariety, 294 residue degree, 364, 403 succeeds, 221 resultant, 151ff., 258ff., 335f. summand, 53 matrix, 152 support, 221 ring, 6ff., 36, 37, 158 supremum, 43 Artinian, see Sylow's first theorem, 5 local, see local ring symmetric group, 3 Noetherian, see Noetherian ring of absolute constants, 204, 271ff. tangent, 309 of constants (of a derivation), 197, 201 space, 310ff., 390 reduced, 243 tensor product, 14 Rees, 280 torsion, 61, 68, 95 semilocal, 235 of elliptic curve, 325, 329, 337, 396 root (of polynomial), 104, 108ff. torsion-free, 61 extension, 136 trace, 14 multiple, 114 bilinear form, 145 tower, 136f., 139 map, 142ff., 254 438 Index

quadratic form, 260 Zariski transcendence cancellation, 206 base, 177 dense, 296f., 390 separating, 268 question, 149 degree, 177, 205, 268, 273 topology, 296f., 299, 387, 390 transcendental, 104, 177 zero (of a ), 304 extension, (purely), 146, 178, 200, 266 zero set, 290 of rank 1, 147 Zorn's lemma, 2 number, 247ff. transfinite induction, 1 transposition, 3 transversal, 3 triangle inequality, 340

UFD, 9, 10, 189, 190, 211, 272, 331, 332f. 354, 404 ultrafilter, 27 ultraproduct, 27 unique factorization domain, see UFD unit, 349 group, see group of units universal, 227f., 279 universal algebra, 12, 25, 36, 45 universal mapping problem, 227 unramified, 364f., 369f. valuation 347ff., 366ff., 374 discrete, 366ff., 402f. equivalent, 347 on Dedekind domain, 374f. p-adic, 337, 347 ring, 350ff., 359ff., 361ff. discrete, 403 of a valuation, 349 value group, 347 Vandermonde matrix, 11 variety, 308 affine, 294 projective, 304ff., 388 quasi-projective, 308

Weierstrass form, see elliptic curve Weierstrass function, 329, 394 Witt cancellation theorem, 22 decomposition theorem, 22 index, 23 ring, 30 word, 221