Shafarevich I.R., Kostrikin A.I. (ed.) — Basic Notions of Algebra :: Электронная библиотека попечительского совета мехмата МГУ

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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..."

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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

Предметный указатель

Fundamental theorem of projective geometry      85
Galileo Galilei      6—7 99
Galileo — Newton group      99—100
Galois extension      99 178 225
Galois group      178
Galois theory      50 102 177 240
Galois, E.      90 102 177 179 239 240 242
Gauss' method (row and column operations)      43 237
Gauss, C.F.      23 43 94 239 240
Gaussian integers      23
General linear group GL(n, K), GL(K)      96 99 115 147 150 160 161 169—170 183 195 232
General linear group, Lie algebra $\mathfrak{g}\mathfrak{l}(n, K)$       190 195
Generalised Cay ley algebra      201
Generalised quaternion algebra      93—95 237
Generators and relations      69 101—102 108 122—123 134—138 139
Generators of a group      101—102 104
Generators of a module      36 42
Generators of algebra or ring      45 68
Generic equation      181
Geometric construction of algebraic operations      13 50 85 242
Geranium      130
Germ of a function      27
Gordan, P.A.      173
Gorenstein, D.      240
Graded algebra or ring      46 63 68
Green, G.      217
Grothendieck, A.      241
Group      100 see knot Lie Lorentz orthogonal representation symmetry transformation unitary
Group $GL(n, \mathbb{F}_q)$       102 124 150
Group algebra or ring $\mathbb{Z}[G]$ , K [G]      64 75 163 222 237
Group character      162 167
Group cohomology       222—225
Group homomorphism      104
Group object in a category      209—210
Group of algebraic type      159
Group of an integral ternary quadratic form      132
Group of automorphisms of free module GL(n, A)      99 236
Group of extensions       103 219—220
Group of finite length      153
Group of knot      138
Group of motions      96 97 107 156
Group of orders $\le$       10—152
Group of rotations of 3-space SO(3)      112 140—142 195
Group, defined by relations      101—102 108 122—123 134—138 139
Group, generated by reflections      119 122—124 242
Hadamard, J.S.      242 243
Hamilton, W.R.      65 174
Hasse — Brauer — Noether theorem (on division algebras over $\mathbb{Q}$ )      95
Hasse's theorem (on division algebra over $\mathbb{Q}$ )      95
Hasse's theorem (on division algebra over $\mathb{Q}_p$ )      93
Hasse, H.      60 93 95
Haussdorff, F.      125
Heisenberg, W.K.      185
Helmholtz — Lie theorem      168—169 243
Hermite, C.      80
Hermitian scalar product      80 168
Higman's theorem      136
Higman, G.      136
Hilbert basis theorem      45
Hilbert, D.      8 45 80 184 239 242
Hilton, P.J.      241
Hirzebruch, F.E.P.      241 243
Hochschild, G.P.      240
Holder, O.      78 154
Hom( , A) and Hom(A, )      208 209
Homeomorphism problem for manifolds      138
Homology , ,       102 208 213—216
Homomorphic image (= quotient)      36 42 45 68
Homomorphism of families of vector spaces      40 230
Homomorphism of groups      104
Homomorphism of modules      36 74
Homomorphism of rings or algebras      24 28 31 63 192
Homomorphism of sheaves      225
Homomorphisms theorem      29 36 69 107
Homotopy theory      102 136 205 208—212 215 219 231—232 241
Hopf algebra      166
Huppert, B.      240
Hurwitz, A.      243
Ideal class group of a ring Cl A      36 61 102—103 235
Ideal of a commutative ring      26 28 32 36
Ideal of a Lie algebra      192
Ideal, generated by a system of elements      36 69
Ideal, left-, right-, two-sided-      68—69 73 83—84
Identity element      11 100
Identity morphism in a category      204
Identity problem (= word problem) for groups      135—136
Image of a homomorphism Im f      25 36 63 74 106 226
Imaginary part of a quaternion Im(q)      65
Incidence axioms      8—9 84
INDEX      249—258
Index of a subgroup (G: H)      106 109
Index of an elliptic operator Ind      2—234
Index theorem      234 243
Infinite group      64 124
Infinite-dimensional representation      80 176—177
Infinitesimal      50—51 55 192
Instantaneous angular velocity      195
Integral domain      19
Integral of a differential form      216
Integral of motion      98
Integral over a group I(f)      168
Invariant differential form      142 168 176
Invariant Hermitian scalar product      80 148 165 168
Invariant of a division ring (algebra) $\mu_p(D)$       93—95 104
Invariant of a group      183 185
Invariant quadratic form      80 114 115
Invariant Riemannian metric      142 198
Invariant subspace      75 76 162 175
Invariant theory      160 183—184
Invariant vector field      142 193—194
Inverse $x^{-1}$       11 66 72 100 134 163 166
Inverse quaternion $q^{-1}$       65 66
Invertible element      22
Involution of rings *      67 71 191
Irreducible polynomial      14 22
Irreducible representation (= simple module)      77 87 88 162 163 170 176
Isomorphism in a category      207
Isomorphism of fields and rings      13 17 24 29 63
Isomorphism of group actions      105
Isomorphism of groups      101
Isomorphism of ideals and modules      35 36
Isomorphism of Lie algebras      191
Isomorphism problem for groups      135—136 139
Isosahedral group $Y \cong \mathfrak{U}_5$       111—112 123
Isosahedron      110—112 158
Isotopic spin (isospin)      172—174 185—187
Jacobi identity      190
Jacobi, C.G.J.      190 191
Jets      55
Jordan normal form      43 80 103
Jordan — Hoelder theorem      78 154
Jordan's theorem on finite subgroups of $GL(n, \mathbb{Z})$       119 124 126
Jordan's theorem on finite subgroups of O(n)      118
Jordan, C.      43 78 80 118 119 154 239 240
K-theory      230—239
K-theory, of a field       237
K-theory, $K_n(\mathbb{Z})$       238—239
K-theory, K(A), $\tilde{K}(A)$ , ,       235—236
K-theory, K(X), $\tilde{K}(X)$ , $\tilde{K}^n(X)$       231
Kaplansky, I.      243
Kernel of homomorphism Ker f      26 63 68 106
Kernel of integral operator      38
Kernel of morphism of sheaves      226
Kirillov, A.A.      240
Klein, F.      105 239 240 241 243
Knot group      138—139
Koebe, P.      131
Kronecker, L.      239 241
Kurosh, A.G.      240
Lagrange's theorem (on sums of four squares)      66

Lagrange, J.L.      23 66 98 240
Laplace, P.S.      72
Lattice $C \subset \mathbb{R}^n$       115 126 143
Laurent series      16 21 57 59
Laurent, P.A.      16 33
Left coset, left ideal, left invariant, left regular      see coset ideal invariant regular
Legendre's theorem (on rational solutions of )      60 94 132
Legendre, A.M.      60
Length      77—78 79 153—154
Lie algebra of a Lie group $\mathscr{L}(G)$       194—197
Lie algebra or ring      188—199
Lie group      125 140 142 143—150 192 210 240 see orthogonal unitary special spinor
Lie subgroup      143
Lie theory      192—199 206 240
Lie's theorem (on the Lie algebra of a Lie group)      196
Lie, M.S.      142 168 169 190 192 196 240 243
Linear dependent elements of a field      40
Linear differential operator      19 34
Linear differential operator of order $\le r$       55
Linear map      36
Liouville's theorem (on integrable systems)      143 243
Liouville, J.      143 228 243
Lobachevsky plane      105 131 149 169
Lobachevsky, N.I.      105 130 243
Local Lie group      196
Long exact cohomology sequence      218—219 220—222 227—228 231 236
Long exact cohomology sequence of a subspace      218—219
Loop (= closed path)      136 182 208
Loop space $\Omega X$       208 210
Lorentz group O(3,1), SO(3,1)      100 148—149 176
Lorentz, H.A.      100
MacDonald, I.G.      239 242
MacLane, S.      241
Magnus, W.      240 243
Mal'tsev, A.I.      242
Manifold      40 47 52 55 56 125 131 142 148 182 189 192 201 214 216—217 225 228 229 233
Manin, Yu.I.      70 240 241 242
Matrix algebra or ring ,       63 73 78 83 89 190
Maximal compact subgroup      148 158 224
Maximal ideal      29 32 40 51—52
Merkur'ev, A.S.      243
Meromorphic functions      16 47 181 226 229
Mesons      185
Milnor, J.W.      241
Minimal polynomial      48
Minkowski — Hasse theorem (on rational solutions of quadratic equations)      60
Minkowski, H.      60
Modular group $PSL(2, \mathbb{Z})$       132—133
Module      33—34 74—79
Module of differential forms      34 35
Module of finite type (= finitely generated)      42 44
Module or rank zero      42
Module over $\mathbb{Z}$       35 100 205
Module over a PID      43 235
Module over K[x] corresponding to linear transformation      34 42 43 77 80
Modulus (= absolute value) of a quaternion |q|      65
Momentum      98
Monodromy of a differential equation      115 162
Morphism in a category      204
Motions      96 195
Multiplication in modules      36
Multiplication table (= Cayley table)      101 151
Newton, Sir Isaac      99
Noether, A.E.      44 95 98 184 239 242
Noetherian module or ring      44—46
Nonassociative (division) algebra      199—200
Noncommutative ring      62
Noncommuting polynomial algebra $K\langle x_1,\cdots, x_n\rangle$       68
Nonsingular point of a variety      54
Nonstandard analysis      32
Normal subgroup $N \lhd G$       106 109 194
Normal subgroup of $\mathfrak{S}_n$ and $\mathfrak{U}_n$       109
Normed field      57
Nucleon      185
Number of irreducible representations      88 163 165
Object of a category $Ob(\mathscr{C})$       204
Octahedral group $O \cong\mathscr{S}_4$       111—112 167
octahedron      110—111
Octavions (= Cayley numbers) O      199—200
Odd permutation      109
Opposite (= skew-isomorphic) ring      67 74 78 82
Orbit of an element Gx      100 106
Orbit space (= quotient) G\X      100 105 106 125 126 169
Order of a group element      107
Order of a group |G|      100
Orthogonal groups O(n), SO(n), PSO(n), O(p, q), SO(p,q),       96 107 141 144 145 147 169 171 195
Orthogonal Lie algebras o(n,K), o(p,q)      190—191 195
Orthogonality of characters      164 167 169
Orwell, G.(Blair, E.A.)      31
Ostrowski's theorem (on valuations of Q)      59
Ostrowski, A.      59
p-adic field $\mathbb{Q}_p$       57 59 93 150 183 241
p-group      156
Pappus (Pappos of Alexandria)      85 91 242
Pappus' theorem or axiom      85 91 242
Parity law      100
Partially ordered set      84
Path or loop      136 208
Periodicity theorem in K-theory      232
Permutation group      98 102 108—109 119 179 240
Picard, C.E.      181
Plato (Platon)      110 158
Platonic solids      110—111 158 169
Poincare model of Lobachevsky space (upper half-plane)      105 131
Poincare — Koebe uniformisation theorem      131
Poincare, H.      105 131
Poisson bracket [,]      7 189
Poisson, S.D.      7 189
Polyhedron      110 112 117 214
Polynomial      18
Polynomial convex hull      25
Polynomial function on a curve      21
Polynomial ring A[x], $K[x_1,\cdots,x_n]$       17—19 26 45 55 56—57
Pontryagin duality theorem      171
Pontryagin, L.S.      171 240
Presheaf      225
Prime ideal      31 61
Prime subfield      30 48
Primitive Element Theorem      49
Principal ideal      26 27 36
Principal ideal domain (PID)      26 43 235
Product in a category      202 207
Product in cohomology      217
Product of differential forms      216
Product of fields      32
Product of ideals      27
Product of two modules with values in a third      36—37
Projective limit of rings      55
Projective module      221 235
Projective resolution of a module      221 232
Projective space $\mathbb{P}^n$       84
Projective space axioms      84
Projective space over a division algebra $\mathbb{P}^{n-1}(D)$       84
Puiseux expansion      59
Puiseux, V.A.      59
Purely imaginary quaternion $q \in \mathbb{H}^-$       65 141 171
Quadratic reciprocity      94
quantum mechanics      8 172 174 185
Quark      188
Quasi-algebraically closed field      91
Quaternion $q \in \mathbb{H}$       65 91 146 152 171 199
Quaternion of modules 1 SpU(l)      141 145
Quaternionic projective line $\mathbb{P}^1(\mathbb{H})$       66
Quillen, D.G.      243
Quotient complex      217
Quotient group G/N      107
Quotient in a composition series      77 154
Quotient Lie algebra      192
Quotient module      36 42 75 76
Quotient representation      75 76

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