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Lorenz F., Levy S. — Algebra, Volume I: Fields and Galois Theory
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Название: Algebra, Volume I: Fields and Galois Theory
Авторы: Lorenz F., Levy S.
Аннотация: From Math Reviews: "This is a charming textbook, introducing the reader to the classical parts of algebra. The exposition is admirably clear and lucidly written with only minimal prerequisites from linear algebra. The new concepts are, at least in the first part of the book, defined in the framework of the development of carefully selected problems. Thus, for instance, the transformation of the classical geometrical problems on constructions with ruler and compass in their algebraic setting in the first chapter introduces the reader spontaneously to such fundamental algebraic notions as field extension, the degree of an extension, etc... The book ends with an appendix containing exercises and notes on the previous parts of the book. However, brief historical comments and suggestions for further reading are also scattered through the text."
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Рубрика: Математика /
Статус предметного указателя: Готов указатель с номерами страниц
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Издание: 1st edition
Год издания: 2005
Количество страниц: 309
Добавлена в каталог: 01.06.2008
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Предметный указатель
, transcendence of 15 52 203
-adic valuation 40 47
17-gon 52 258
Abel, Niels Henrik (1802—1829) 154 166 178 244
Abelian see also under “Galois group”
Abelian extension 151
Abelian field extension 152
Abelian group 149
Abelian group, elementary 268
Abelian group, finite 154 160
Abelian group, free 161
Abelian number field 259
Adjunction 6
Adjunction of a square root 7
Affine algebraic set 217
Affine coordinate ring 224
Affine k-algebra 224
Affine subgroup of 187
Affine variety 222
Alfes, Rainer ii
Algebra over K 11
Algebraic closure in an extension 19
Algebraic closure, absolute 57
Algebraic extension 17
Algebraic geometry 217
Algebraic independence of field homomorphisms 261
Algebraic K-set 217
Algebraic number 15
Algebraic number field 259 275
Algebraic number theory 113
Algebraic over K 15
Algebraically (in)dependent 209
Algebraically closed 56 234
Alternating group 182
angle see “Trisection”
Artin, Emil (1898—1962) 55 113 118 120 147 261 268
Associated element 34
Automorphism, group of a field extension 65 76 246
Automorphism, inner 96
Axiom of Choice 39
Bertrand, Joseph Louis Francois (1822—1900) 271
Bertrand’s Theorem and Postulate 271
Bezout ring 235
Bezout, Etienne (1730—1783) 235
BI-Wissenschaftsverlag ii
Bilinear form 134
Bilinear map 150
Binomial equation 143
Borevich, Zenon I. (1924—1995) 260
Brandis, Albrecht 232
Capelli, Alfredo (1855—1910) 267
Cardano, Geronimo (1501—1576) 165 185
Carlitz, Leonard (1907—1999) 260
Castelnuovo, Guido (1865—1952) 280
Casus irreducibilis 186
Cauchy, Augustin (1789—1857) 101 154
Cayley, Arthur (1821—1895) 94
Center of a group 97
Central subring 191
centralizer 97
Chain (in ordered set) 64
CHARACTER 149
Characteristic of a field 30
Characteristic subgroup 186 254
Characteristic-simple group 254
Chebyshev, Pafnuti L. (1821—1894) 271
Chinese remainder theorem 42 88
Choice, axiom of 39
circle see also “Quadrature”
Circle, n-section of 3 9 51 104 258
Circle, rational parametrization 279
CM-field 275
Coers, H. ii
Cohen, I.S. 274
Commutative ring with unity 33 133
Commutator subgroup 268
compass see “Ruler and compass”
Complex multiplication field 275
Composite of fields 20
Conjugate, group element 97
Conjugate, over K 65
Conjugate, subgroup 98
Constructibility see “Ruler and compass”
Constructible number 2 5
Content of a polynomial 48
Coset 95
Crossed homomorphism 263
cube see under “Doubling”
Cubic equation 165 185
CYCLE 179
Cyclic extension 144
Cyclic Galois group 81 92 139 143
Cyclic permutation 179
Cyclotomic field 259
Cyclotomic polynomial 90 91 250
Daldrop, Hans ii
Davenport, Harold (1907—1969) 260
Decomposition see “Factorization”
Dedekind reduction principle 190
Dedekind, Richard (1831—1916) i 10 55 117 190 200 242
Degree of a field extension 11
Degree of an element 17
Degree, formula 12
Delian problem 2 27
Derivative 69
Descent lemma 228 281
Determinant of group 262
Deuring, Max (1907—1984) 123
Dimension of a ring 225
Dimension of an algebraic set 224
Directly indecomposable modules 123
Directory of primes 39
Dirichlet, Peter Gustav Lejeune (1805—1859) 251
Discriminant of algebraic number field 275
Discriminant of basis 274
Discriminant of polynomial 81 165 183 272 274
Divisible, divisor 33 (see also “Factorization”)
Double coset 98
Doubling of the cube 2 9 27
Drinfeld, G. I. 203
Dual of group 149
Eisenstein criterion 50 53 73 80 189
Eisenstein, Gotthold (1823—1852) 50 241
Elementary Abelian group 268
Elementary symmetric functions 174 177 195 273
Elliptic curve 236 279
Engesser, Hermann ii
Epkenhans, Hans ii
Euclid (ca. 365—300 B.C.) 26 36 38 236 252
Euclidean algorithm 38 236
Euclidean Domain 36 38 238
Euclidean valuation 36 38
Euler -function see “Totient”
Euler — Lagrange Theorem 95
Euler, Leonhard (1707—1783) 27 105 108 113
Euler’s criterion 108
Exact sequence 150
Exponent of a group 155
Exponent of an abelian extension 152
Exponent of an abelian group 149
Extension, algebraic 17
Extension, field 6 15
Extension, finite 17 193
Extension, integral 193
Extension, normal 59
Extension, ring 191
Extension, simple 21 31
Extension, solvable by radicals 166
Extension, transcendental 17
Factorization 37 39 58
Faltings, Gerd (1954— ) 279
Fermat number 258
Fermat, Pierre de (1607—1665) 105 236 238
Ferro, Scipione (1465—1526) 165
Field of algebraic numbers 19
Field of constructible numbers 5
Field of fractions see “Fraction field”
Field of n-th roots of unity 88
Field of rational functions 29 53 173 190
Field, algebraically closed 56 234
Field, complex multiplication 275
Field, cyclotomic 259
Field, extension see under “Extension”
Field, finite 71 83 86 92 260
Field, fixed 75
Field, Hilbertian 190
Field, intermediate 19
Field, perfect 73
Field, prime 31
Field, Pythagorean 263
Field, quadratic 259
Field, quadratically closed 5
Field, splitting 58
Finite abelian group 154
Finite field 71 83 86 92 260
Finite field extension 17
Finite ring extension 193
Fixed field 75
Formal fractions 235
Four-group 182
Fraction field 28 235
Fractions, ring of 235
Free abelian 161
Frenicle de Bessy (1605—1675) 105
Frobenius automorphism 92
Frobenius, Georg (1849—1917) 92 200 255 257 259 262
Fundamental Homomorphism Theorem 23
Fundamental theorem of algebra 27 36 189
Fundamental Theorem of Galois Theory 79 128
Fundamental Theorem on Symmetric Functions 177 195 273
Galois, Evariste (1811—1832) i 76 166 187 249
Galois, Evariste (1811—1832), extension 76 246
Galois, Evariste (1811—1832), group 76 79 cyclic abelian”)
Galois, Evariste (1811—1832), group of certain polynomials 189
Galois, Evariste (1811—1832), group, abelian 151 259
Galois, Evariste (1811—1832), group, cyclic 81 92 197
Galois, Evariste (1811—1832), group, finite 263
Galois, Evariste (1811—1832), group, modulo primes 197
Galois, Evariste (1811—1832), group, solvable 173
Galois, Evariste (1811—1832), theory 55 76 79 115 128
Galois, Evariste (1811—1832), theory, inverse problem 178 202
Gauss, Carl-Friedrich (1777—1855) 1 22 27 45 49 52 89 104 113
Gaussian integers 238
Gaussian periods 257
Gaussian sums 110
Gauss’s Lemma 49 196
Gauss’s theorem 195
GCD 34
General polynomial 175
Generators 161
Goethe, Johann Wolfgang von (1749—1832) i
Going up theorem 274
Greatest common divisor 34
Group see also “Abelian finite Galois”
Group, action 93
Group, action, transitive 80
Group, algebra 60
Group, determinant 262
Group, theory 93
Hartshorne, Robin 217 280
Heptagon, regular 51 52
Heptakaidecagon 52 258
Hermite, Charles (1822—1901) 52 207
Hilbert, David (1862—1943) 140 190 263
Hilbert, David (1862—1943), Basis Theorem 221 280 281
Hilbert, David (1862—1943), irreducibility theorem 190 201
Hilbert, David (1862—1943), Nullstellensatz 219 280 281
Hilbert, David (1862—1943), Theorem Ninety 140 263
Hilbertian field 190
Homogeneous polynomial 242
Homomorphism of field extensions 55
Homomorphism of K-algebras 55
Homomorphism, crossed 263
Homomorphism, theorem 23
Huppert, Bertram (1927— ) 257
Hyperelliptic curve 279
Hypersurface 229
Ideal 22
Ideal of a set in 217
Ideal, maximal 41
Ideal, prime 41
Ideal, principal 34
Ideal, product of 42 43
Ideal, reduced 219
Ideal, sum of 43
Index of a subgroup 95
Infinite Galois extensions 128
Inner automorphism 96
Inseparable 66
Inseparable degree 73 246
Inseparable purely 71
Integer, Gaussian 238
Integral, basis 275
Integral, closure 194
Integral, domain 16 41
Integral, over 191
Integral, ring extension 193
Integrality equation 191
Integrally closed 194 195
Intermediate field 19
Inverse Galois theory 178 202
Inversion formula 264
Irreducibility criterion 241
Irreducible 37
Irreducible algebraic set 223
Irreducible K-component 223
Irreducible polynomial 26 48
Irreducible radicals 171
Ischebeck, Friedrich G. 280
Jacobi symbol 112
Jacobi, Carl Gustav Jacob (1804—1851) 112
Jensen, Christian U. 202
k-algebra 242
K-conjugate 65
K-homomorphism 55
Klein, Felix (1849—1925) 182
Kronecker, Leopold (1823—1891) 27 58 113 188 230 234 259 275 282
Krull topology 128
Krull, Wolfgang (1899—1971) 123 128 225 228 274 281
Krull’s descent lemma 228 281
Kummer extension 152
Kummer theory 153 265 268
Kummer, Eduard (1810—1893) 152 276
Kummer’s Lemma 276
Lagrange resolvent 145
Lagrange, Joseph Louis (1736—1813) 27 95 145
Lang, Serge (1927— ) 190
LCM, least common multiple 34
Ledet, Arne 202
Legendre symbol 107 149
Legendre, Adrien Marie (1752—1833) 107 113
Leibniz, Gottfried Wilhelm (1646—1716) 259 262
Length of cycle 179
Length of orbit 94
Levy, Silvio ii
Lindemann, Ferdinand (1852—1939) 15 52 207
Linear algebra i 11 133 242
Linear independence of field homomorphisms 117
Local ring 237 274
localization 235 237
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