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Lorenz F., Levy S. — Algebra, Volume I: Fields and Galois Theory
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."


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 2005

Количество страниц: 309

Добавлена в каталог: 01.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\pi$, transcendence of      15 52 203
$\pi$-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 $S(\mathbb{F}_p)$      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 $\varphi$-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 $C^n$      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 $\mathbb{R}$      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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