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Shafarevich I.R., Kostrikin A.I. (ed.) — Basic Notions of Algebra
Shafarevich I.R., Kostrikin A.I. (ed.) — Basic Notions of Algebra

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Название: Basic Notions of Algebra

Авторы: Shafarevich I.R., Kostrikin A.I. (ed.)

Аннотация:

From the reviews: "... This is one of the few mathematical books, the reviewer has read from cover to cover ...The main merit is that nearly on every page you will find some unexpected insights... " Zentralblatt für Mathematik "... There are few proofs in full, but there is an exhilarating combination of sureness of foot and lightness of touch in the exposition... which transports the reader effortlessly across the whole spectrum of algebra...Shafarevich's book - which reads as comfortably as an extended essay - breathes life into the skeleton and will be of interest to many classes of readers; certainly beginning postgraduate students would gain a most valuable perspective from it but... both the adventurous undergraduate and the established professional mathematician will find a lot to enjoy..."


Язык: en

Рубрика: Математика/Алгебра/Абстрактная алгебра/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 258

Добавлена в каталог: 05.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$Ext_R(L,M)$      103 219—220
$\mathbb{Z}$-module (= Abelian group)      100 205
$\mathbb{Z}/2$-grading      70
$\Omega^-$-hyperon      188
'Coordinatisation'      6—11 14 17 21 24—25 49 85 102 160 240
a.c.c. (= ascending chain condition)      41 44
Abel, N.H.      35 100 240
Abelian group      35 42 43 100 155 163 170 203 213
Abelian group, Lie algebra or ring      192
Absolute value (= modulus) of a quaternion |q|      65
Abstract group      100—102 104—108 151 240
Action of a group      104—105 132 192
Adele group      151
Adjoint action      105
Algebra      4—248 63 74
Algebra of differential forms      71
Algebra over a ring      63 203—204
Algebraic element of a field      48
Algebraic extension      48
Algebraic integers      61
Algebraic matrix group (= linear algebraic group)      150 155 157 159 181 224
Algebraic number field      31 61 103 235
Algebraically closed field      29 49 88 91
Algebraically dependent elements      47
Alternating group $\mathfrak{U}_k$      109 157 159
Alternative ring      200
angular momentum      98 198 242
Arithmetic genus      229
Arithmetic group or subgroup      132 133 150—151 224 239
Arnol'd, V.I.      143 242 243
Associative law      11 100 188 190 196 199 223
Atiyah — Singer index theorem      234
Atiyah, Sir Michael      234 239 240 241 242
Automorphic function      131 177
Automorphism      99
Automorphism of an extension      99 177—178
Axiomatic projective geometry      84 201
Banach, S.      38
Baobab tree      130
Baryons      185 187
Basis of free module      35
Bianchi, L.      172
Bieberbach, L.      126
Bogolyubov, N.N.      243
Boole, G,      22
Boolean ring      21—22 25 241
Borel, A.      240 243
Borel, E.      56
Bose, S.N.      162
Bosons      162
Boundary      213 215 217
Bouquet of topological spaces $X \vee Y$      137 207
Bourbaki, N,      239 242 243
Braid group $\Sigma_n$      139 243
Brauer group Br K      93—96 103—104 237
Brauer, R.D.      95 103
Bravais lattices and groups      115—118 127
Bravais, A.      115
Brouwer fixed point theorem      215
Brouwer, L.E.J.      215
Brown, K.S.      241
Burnside's theorems      88—89 163 167
Burnside, W.S.      88 89 240
Campanula      130
Canonical homomorphism to quotient      28 36 107
Cartan, E.      197
Cartan, H.      241
Categories $Al, Let, Mod_r, Top, Top_0, Hot, Hot_0$      205 215 219
Category      202 204 225 241
Category of formal groups      206
Cauchy, A.L.      57 72
Cayley numbers (= octavions) $\mathbb{O}$      199—200
Cayley table (multiplication table) of a finite group      101 151
Cayley — Klein model of Lobachevsky space      105 131 132 149
Cayley, A.      101 105 200 241
Central algebra      90 103—104
Centre of an algebra      63 90
Chain conditions      41 44 77—78
Chain, complex $\{C_n, \partial_n\}$      213
Chandler, B.      240 243
CHARACTER      89 163—165 167 173
Character of SU(2)      172—174 186
Character, group G      163—164 170
Characteristic of a field      31
Che valley, C.      91 240 243
Chevalley's theorem (finite fields are quasi-algebraically closed)      91
Class field theory      95
Class group:      see ideal class group
Class number      36
Classical groups      144—150 157—158 174—176
Clebsch — Gordan formula (for representations of SU(2))      173 186
Clebsch, R.F.A.      173
Clifford algebra C(L)      70 71 72 73 87 145 149 242
Clifford, W.K.      71 242
Coboundary homomorphism      213
Cochain complex      213
Cohomology $H^n(K)$, $H^n(K, A)$, $H^n(X, A)$      208 213 214 216
Cohomology ring $H*(X, \mathbb{R})$ $H*(X,\mathbb{C})$      217 234 see sheaf long
Commutative diagram      202
Commutative group:      see Abelian group
Commutative Lie ring      192
Commutative ring      17—23
Commutative ring of continuous function $\mathscr{I}(X)$      20 24 234—235
Commutative ring of differential operators $\mathbb{R}[\frak{\partial}{\partial x_1}, \cdots, \frak{\partial}{\partial x_n}]$, $\mathbb{C}[\frak{\partial}{\partial x_1}, \cdots, \frak{\partial}{\partial x_n}]$      19 33 34 42 71 189
Commutative ring of finite type      45
Commutative ring of germs of $C^{\infty}$ functions $\mathscr{E}$      27 44 52 56
Commutative ring of germs of holomorphic functions $\mathscrO}_n$      21 22 26
Commutative ring of integers $\mathscr{Z}$      17 22 26 28 31 54 56
Commutative ring of integers of an algebraic number field      31 61 103 235
Commutative ring of p-adic integers $\mathbb{Z}_p$      56—57
Commutative ring of polynomial functions on a curve K[C]      21 33 40 45 56 see polynomial
Commutator in a group G'      155—156 158—159
Commutator of differential operators (derivations)      189
Commutator [ , ]      189 194
Commuting differential operators      33
Compact Lie groups      143—147 167—174
Complement of a normal subgroup      223
Completion $\hat{A}, \hat{L}$      55—57 59
COMPLEX      214
Complex Lie groups      147—150 174—177
Complex manifold      47 147 148 226 228 229
Complex torus      48 143
Complexification of vector space      38
Composite of loops      136 210 211
Composition series      77—78 154
Congruence mod I      28
Conjugacy classes      106 109 165
Conjugate element or subgroup      106 109
Conjugate quaternion q      65 93 199
Conservation of momentum      98
Continuous geometry      87 242
Contragredient (= dual) representation p      165 166 172 188
Contravariant      see functor tensor
Convolution      64
Coordinate ring K[C]      21 33 40 45 56
Coset (= residue class) mod I      28 36
Coset (left- or right-)      105—106
Covariant:      see functor tensor
Crystal (or crystallographic) class      115 116 117 124 242
Crystallographic group      97—98 126—131 155
Crystallographic group in Lobachevsky plane      130—131
cube      110—111 158
Curvature tensor      172
CYCLE      215
Cycle type of a permutation      109
Cyclic group or subgroup      107 108 110 111 180
Cyclic module      43
de Fermat, P.      23
de Rham cohomology $H'_{DR}(X)$      216—217
de Rham's theorem      217
de Rham, G-W.      216 217 241
Dedekind, J.R.W.      239 241
Defining relations      see generators and relations
Degenerate kernel      38
Degree of an extension [L : K]      48 99 182—183
Delone, B.N.      242
Demushkin, S.P.      183 243
Derivation      53 189
Derived functor      see long exact cohomology sequence
Desargues' theorem or axiom      85—86 91 201 242
Desargues, G.      85 242
Descartes, R.      6
Deuring, M.F.      239 242
Diagram      202
Dictionary of quantum mechanics      8 38 185 186
Differential automorphism      181
Differential form      34 35 36 53 216 226
Differential Galois group      181
Differential of a complex      213
Differential operator      19
Differential operator on a manifold      53 193
Differential operator on a vector bundle      233
Dihedral group $D_n$      110 111 112 118 123 152
DIMENSION      41 206
Dimension of a Lie algebra dim if      191
Dimension of a variety dim C      47
Dirac equation      71 242
Dirac, P.A.M.      71 162 242
Direct product of groups      108 153 170 207
Direct sum of fields      83 163
Direct sum of modules      34 79 207
Direct sum of representations      76 88 89 162
Direct sum of rings      20 81 207
Dirichlet character      165
Dirichlet, P.G.L.      165
Discrete (= discontinuous) group      124—125
Discrete group in Lobachevsky space      131—133 243
Discrete series      177
Discrete subgroup of $\mathbb{R}^n$      125—128
Discrete transformation group      125 138
Divisibility theory in a ring      22 23
Division algebra      66 72 77 78 83—86 88 90—96 200 237
Division algebra over $\mathbb{R}$      91 201 233 239
Divisor on a Riemann surface      226 229
dodecahedron      110—111 158
Dold,A.E.      241
Dual category $\mathscr{L}*$      206
Dual module M*      39
Dual polyhedron      110—111
Dyson, F.J.      243
E.Noether's theorem (on symmetries and conservation laws)      98
Egyptian craftsmen      128 242—243
Eilenberg, S.      241
Einstein, A.      162
Elementary particles      158 162 174 185—188
Elliptic functions      126
Elliptic operator      233
Empty set      84
Endomorphism ring End^M      62 63 67 69 75 79 166
Engel, F.      240
Equivalent extensions      103
Equivalent functors      209
Equivalent lattices      115
Equivalent representations      76
Error-correcting code      30
Euclid (Euklides)      22 35 158
Euclidean algorithm      22 43 237
Euler characteristic      224
Euler characteristic of a group      224
Euler characteristic of a sheaf $\chi(X, \mathscr{F})$      228—230 234
Euler equations for rigid body motion      198
Euler substitutions      16
Euler, L.      16 23 65 93 141 198 224 238
Evaluation homomorphism      24 27 51
Even Clifford algebra $C^0(L)$      71 72 73 145 149
Even permutation      109 112
Exact sequence      218 226—228 231 236
Exceptional simple Lie groups $E_6, $E_7$, $E_8$, $G_2$, $F_4$      158 197 201
Extension of finite type      46
Extension of groups      154—155 223—224
Extension of modules      103 219—220
Extension of the group field      92 198
Exterior algebra of a module or vector space $\bigwedge M$      70—71
Exterior power of a module $\bigwedge 'M$      39
Exterior product $x \wedge y$      39
Factor group      see quotient group
Factorisation      22 23 26—27 61
Family of vector space $\mathscr{E} \to X$      40 230
Fano's axiom in projective geometry      86
Fano, G.      86
Fedorov, E.S.      242
Fermi, E.      162
fermions      162
Feynman, R.P.      242
Field      11—17 26 29 72 235
Field of (formal) Laurent series K((t))      16—17 21 56
Field of fractions of a ring      19 31
Field of meromorphic functions Ji(X)      16 47
Field of rational functions K(x), $K(x_1, \cdots, x_n)$      13 19 47 49 58 180
Field of trigonometric functions      16
Field, $\mathbb{Q}, \mathbb{U}, \mathbb{C}$      12 29 31 46 59 94
Field, extension L/K      12 29 46 49
Final object in a category      210
Finite abelian group      42 43
Finite dimension      41—45 191
Finite extension L/K      48—49 63 103 177
Finite field $\mathbb{F}_q$      10 12 29 30 46 48 86 91 124 159 177 238 242
Finite geometry      9 86 91
Finite group      64 100—101 106 107 108—124
Finite group of algebraic type      159
Finite group of fractional-linear transformations      114
Finite group of orthogonal transformations      110 112—114 115
Finite length      77—78 153—154
Finite rank      41 63
Finite reflection groups      119 122—124
Finite sheeted cover      182—183
Finite simple groups      159
Finitely generated abelian group      44 153
Finitely generated algebra or ring over A (= of finite type)      45
Finitely generated extension      46—49
Finitely generated group      135
Finitely generated module (= of finite type)      42
Finitely presented group      135—136
First order linear differential operators      53 55 62 189
Flabby (= flasque) sheaf      227—228
Flabby resolution of a sheaf      228
Flag      110 168—169
Formal group (law)      197 206
Formal power series ring K[t], $K[x_1, \cdots, x_n]$      21 56
Fourier coefficients      164
Fourier series      25 143 170 177
Fourier transform as isomorphism of modules      35
Fourier, J-B-J.      25 35 170
Fredholm operators      69 234
Fredholm, E.I.      69
Free action of a group      125 138
Free generators (= basis)      35
Free groups $\mathscr{I}_n$      134 135 137 138 203
Free module      34—35 41 221
Free product of groups      203 207
Freudenthal, H.      241
Frobenius' theorem (on division algebras over $\mathbb{R}$)      91
Frobenius, F.      91 240
Function field K(Q      14 31 47 49 91
Functional view of a ring      18 24 31 40 53—54 61 234—235 238 241
functor      207 208 215 219 226 231
Functor $Vec(X)$      230
Fundamental domain for a discrete group      116 125
Fundamental group $\pi(X)$, \pi_1(X)$, \pi_1(X, x_0)$      136—138 142 182 208
Fundamental Theorem of Galois Theory      179—182
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