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Index

Acyclic relation, 28 Additive monoid, 35 Adjacent words, 249 Adjoint matrix, 71 Admits splitting fields, 203 Algebra, 232 Algebraic, 139

numbers, 189 Algebraically

closed field, 152 (in)dependent, 145

Algorithm, 2, 30 Apartness, 8, 30

domain, 64 Approximation to the N-ba1l, 293 Archimedean

valuation, 288, 290, 291 Artin- Rees lemma, 200 Ascending chain condition, 23 Associated prime ideal, 217 Associates, 108 Automorphism, 18 Axiom of choice, 14

countable, 15 dependent, 15 unique (nonchoice), 10, 31 world's simplest, 15, 33

Azumaya theorems, 97, 98

Basis of a free module, 56 of a free group, 249 of a normed vector space, 300

Belongs to, 208 8ezout domain, 115 Bifunctor, 90 Bijecti on, 10 Bilinear map, 89 Bimodule, 53 Binary sequence, 4 Bounded

cancellation monoid 109 depth, 24 discrete domain, 109 element, 109

339

in number, 11 metric space, 292 module, 211 set, 11

Bounding constant, 287 Brouwerian example, 5 Burnside'S theorem, 237

Cancellation monoid, 108 Cardinali ty, 10 Cartesian product, 9 Category, 16 Categorical

coproduct, 20 product, 19

Cauchy sequence, 48 Cayley-Hamilton theorem, 72 Center, 232 Centralizer, 233 Chain, 22 Character of a monoid, 172 Characteristic

function, 13 of a ring, 45 polynomial, 71

Chinese remainder theorem, 329 Church-Rosser property, 54, 250 Cofactor, 70 Coherent, 82 Column operations, 66 Commutative

diagram, 93 monoid, 35 ring, 41

Companion matrix, 136 Complement, 9 Complementary summand, 57 Complete metric space, 51 Completion

of a valued field, 295 of a metric space, 51

Complex field, 302 numbers, 50

Composition series, 221

340

Computational interpretation, 2 Condition P, 187 Conjugate, 261, 262 Consistent

basis function, 205 inequali ty, 8

Content, 123 Converges, 51

to zero, 48 Coproduct, 20 Coset, 39 Cotransitive inequality, 8 Cotransitivity of a < b, 49 Countable

axiom of choice, 15 set, 11, 32

Cycle of a permutation, 37 Cyclic

group, 37 module, 56

Cyclically reduced word, 261

DH-ring, 121 Decidable relation, 7 Dedekind-Hasse map, 118 Dedekind-infinite set, 14 Dedekind

domain, 326 set of valuations, 326

Degree of a polynomial, 60, 61 Denial

field, 45 inequality, 8

Dense subspace, 51 Density theorem, 235, 236 Dependent vectors, 68 Depth, 23, 24 Derivative (formal), 154 Descending chain condition, 23, 28 Detachable

ideals and submodules, 83 subset, 9

Determinant, 69, 70 Diagonal matrix, 128 Diagonalizable, 137 Difference relation, 12, 30 Dimension of a vector space, 67 Direct

limit, 20, 165 sum, 53, 54 summand, 57 product, 53

Discrete set, 9 valuation, 293

Disjoint union, 18

Distributive lattice, 21, 24 Divides, 63, 108, 113 Divisible

abelian group, 269 hull, 271

Division algebra, 233 algorithm, 61 ring, 41

Divisor chain condition, 110 Domain, 42 Dual category, 20

Eigenvalue and eigenvector, 136 Eisenstein criterion, 117 Elementary

divisors, 134 matrix, 66

Index

row and column operations, 66 symmetric polynomials, 73

Embedded prime ideal, 217 Endomorphism, 18

ring, 41 Epimorphism, 18 Equality relation, 7 Equivalent

matrices, 128 norms, 299 valuations, 288

Essential subgroup, 271 Euclidean

algori thm, 63 domain, 118 map, 118

Even permutation, 37 Exact sequence, 90 Exchange property, 145 Existence, I, 2 Extensional, 7, 9

strongly, 10

Factor, 63 Factorial

domain, 114 field, 176

Faithful module or representation, 52, 232

Family, 11, 18 Field, 42

generated by, 43 norm, 317 of definition, 155 of quotients, 42

Finitary tree, 6, 7 Finite

cardinality, 10, 11

Index

depth, 24 dimensional, 67, 232 family, 11 presentation, 130 rank free module, 56 rank torsion-free group, 265 routine, 2 set, 11, 32 support, 26, 53

Finitely generated, 37, 43, 56 enumerable, 11 presented extension field, 199 presented module, 81 presented ring, 198

Fixed field, 171 Flat module, 92 Fractional ideal, 329 Formal power series, 64, 298 Free

group, 249 module, 56 monoid, 36

FUll subgroup, 269 FUlly

factorial field, 188 Lasker-Noether ring, 224

Function, 9 Functor, 18 Fundamental theorem

of algebra, 191

GCD (domain and monoid), 108 Galois extension and group, 169 Gauss's lemma, 123 Gelfand-Tornheim theorem, 303 General valuation, 287 Greatest common divisor (see GCD) Group, 36

ring, 60

Height (see depth) of a prime ideal, 228 in an abelian p-group, 273 in a torsion-free group, 266

Hensel's lemma, 311, 315 Henselian

context, 309 field, 311

Henselization, 305, 314 Hereditary subset, 24 Heyting field, 42, 77, 96 Hilbert basis theorem, 196 Homomorphism, 18

of algebras, 232 of modules, 52

of monoids, 35 of rings, 43

Hopfian, 80, 85, 257

Ideal, 43, 44, 53 Idempotent, 57 Identity

function, 9 matrix, 65

Image, 10 Impotent ring, 104, 141 Index of a subgroup, 39 Independent

over a local ring, 99 submodules, 53 vectors (discrete case), 68

Inequivalent valuations, 288 Indeterminate, 60 Inequality, 8, 31

on a local ring, 98 Infimum, 21 Infinite set, 11 Inhabited, 30 Injection of ordinals, 28 Integral

closure, 139 domain, 42 element, 139 extension, 139

Integrally closed, 139, 140 Intermediate value theorem, 189 Intersection, 8, 11 Invariant factors, 138 Inverse, 36

functions, 10 Irreducible

element, 109 ideal, 220 module, 235

Irredundant primary decomposition, 217

Isolated primary ideal, 219 Isomorphism, 18

Jacobson radical, 78

341

Jordan canonical form, 106, 136 Jordan-HOlder-Dedekind theorem, 22

I ~ -homomorphism, 167 Kernel, 38, 52 Konig's lemma, 7 Kripke's schema, 32 Kronecker 1, 125 Kronecker 2, 126 Krull

intersection theorem, 200, 201

342

LLPO, 5, 33, 51 LPO, 4 Lagrange interpolation formula, 64 Lasker-Noether

ring, 220 theorem, 227

Last-difference relation, 26 Lattice, 21 Law of excluded middle, 4 Leading coefficient, 61 Least common multiple, 113 Left

ideal, 43, 53 inverse, 36

Length of an abelian group, 274 of a word, 250

Limit, 51 Linearly

independent, 99 independent functions, 172 ordered, 22

Local ring, 85, 96 Localization, 85 Locally (pre)compact, 292, 293 Located subset, 300 LUroths's theorem, 148

Map of partially ordered sets, 20 Maschke's theorem, 242 Markov's principle, 33 Matrix product, 65 Maximal

chain, 22 ideal, 45, 77 left ideal, 79

Metric basis, 299 space, 50

Metrically independent, 299 Minimal

polynomial, 135, 144, 234 prime ideal, 214

Modular lattice, 21 Module, 52, 232

of fractions, 85 Monic polynomial, 61 Monoid, 35

ring, 60 Monomial, 60 Monomorphism, 18 Multiplicative

function, 118 monoid, 35 norm, 118 submonoid, 85

Index

Nakayama Lemma, 78 Newton interpolation formula, 62 Newton's method, 296 Nielsen

set, 253 transformation, 255

Nilpotent element, 78, 141 left ideal, 237

Noether normalization, 198 Noetherian

basis function, 204 module, 80 ring, 80

Nonarchimedean valuation, 288, 290 Nonempty, 7, 8, 30 Nongenerator, 79 Nontrivial

homomorphism, 35 valuation, 288

Nonzero left ideal, 43 Norm, 118, 299 Normal

extension field, 168 subgroup, 38

Normed vector space, 299 Nullstellensatz, 203

Odd permutation, 37 w-bounded set, 14 Omniscience principle, 4 One-to-one function, 10 Onto function, 10 Operation, 31 Opposite ring, 58 Order of a group element, 36 Ordinals, 25

equality of, 28 Grayson, 29

Ostrowski's theorem, 305

p-adic metric on (]), 51 valuation, 292

p-group (abelian), 273 partially ordered set, 20 Perfect

closure, 165 field, 164

Permutation, 37 matrix, 67

Piecewise isomorphic, 22 polynomial, 60 positive real number, 49 Power set, 7 Preimage, 10

Index

Primary decomposition, 216 decomposition theorem, 222 ideal, 208 module, 134

Prime field, 43 ideal, 45 in a cancellation monoid, 109

primitive element, 158 polynomial, 123

principal ideal, 44, 110 domain, 115 theorem, 228, 229

Product of algebras, 239 Projection, 58 Projective

module, 57 set, 16

proper element in a p-group, 279 ideal, 43 subset, 9

Prtifer's theorem, 280 Pseudofactorial field, 295 Pseudonorm, 118 Pure

subgroup, 281 submodule, 84, 92

Purely inseparable, 164 transcendental, 146

Quasi-UFO, 110 factorization theorem, 111 regular, 78

Quaternion algebra, 235 Quotient

of an ideal, 44 group, 38

Radical Jacobson, 78 of an ideal, 44 of an algebra, 237

Ramification index, 319 Rank

function, 34 of a free group, 252 of a free module, 56 of a torsion-free group, 265 relation, 29

Rational canonical form, 136

function, 102 quaternions, 46 root test, 179

Real field, 302 numbers, 48

Recognizable units, 41, 108 Reduced

abelian group, 274 form of a word, 251 length, 253 valuated p-group, 278 word, 251

Reducible word, 251 module, 235

Rees ring, 200 Reflect finitely

generated ideals, 197 Relation, 7 Relatively prime, 63, 108 Remainder theorem, 62 Representation, 52, 232 Representable functor, 20 Residue class

degree, 320 field, 293

Right ideal, 43, 53 inverse, 36

Ring, 41 of fractions, 85 with detachable ideals, 83

Root of multiplicity n, 65 test, 179

Row operations, 66 space, 66

Saturated multiplicative submonoid, 115

Scalar multiplication, 52 Schanuel's trick, 57 Schreier transversal, 257 Schur'S lemma, 237 Seidenberg field, 187 Semirigid class, 267 Separable

closure, 163, 185 element, 155 extension, 155 polynomial, 154

Separably closed in E, 155, 163 factorial field, 176

343

344

Set, 7 Sequence, 11 Simple

module, 235 root, 65

Simply existential statement, 6 presented p-group, 280

Smith normal form, 129 Socle of an algebra, 241 Span operation, 150 Splitting field, 152 Standard inequality, 8 Strictly bounded domain, 121 Strong

approximation theorem, 328 inequality, 99

Strongly extensional function, 10 relatively prime, 63

Subfini te, 32 Subgroup, 37 Submodule, 53 Submonoid, 35 Subring, 42 Subset, 7, 31 Successor, 281 Summand, 57 Support of a cycle, 37 Supremum, 21 Surjective function, 32 Sylow subgroup, 190 Symbolic power, 214 Synunetric

group, 37 inequality, 7 polynomial, 73

Tennenbaum, 204, 231 module, 207

Tensor product, 89 Tight inequality, 8 Torsion, 133, 211

forest, 280 Torsion-free abelian group, 265 Transcendence

basis, 146 degree, 146

Transitive set (relation), 25 Transversal , 257 Trivial

metric, 301 ring, 41 valuation, 292

Type, 266

UFD, 114 U-function, 282 Ulm

function, 278 group, 277 invariant, 278

Ulm's theorem, 279 Union, 8, 11 unique

choice, 10,31

Index

factorization domain (see UFD) interpolation, 62

Unit, 36

valuated p-group, 278 Valuation, 287 Value group, 287 Valued field, 287 Valuation ring, 130 Vandermonde determinant, 73 Variety, 204 vector space, 67 Von Neumann regular ring, 69, 96

weak approximation theorem, 327 Nullstellensatz, 142, 143

wedderburn's theorem, 243, 246 Well

founded, 25 ordered, 25

WLPO (weak LPO), 6 Word, 36

problem, 263 world's simplest

axiom of choice, 15, 33

Zero-divisor, 233 Zippin's theorem, 282