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Kurosh A. — Higher Algebra
Kurosh A. — Higher Algebra



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Название: Higher Algebra

Автор: Kurosh A.

Аннотация:

The present textbook on higher algebra is designed for the mathematics departments of universities. It covers the fundamentals of linear algebra, the primary aim of which is the study of arbitrary systems of first-degree equations (linear equations) and the algebra of polynomials, where the emphasis Is on equations In one unknown but of arbitrary degree. The attractive feature of this text is the mathematically rigorous presentation of the material coupled with Its excellence as an educational tool. Its value Is enhanced by the abundance of examples and problems, many of which have been worked out in great detail. The book has gone through nine editions In Russian and has been translated into a number of other languages. For many years It has enjoyed great popularity among students and teachers alike.


Язык: en

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1975

Количество страниц: 428

Добавлена в каталог: 13.02.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Factor groups      394 395 396
Factor groups, examples of      397
Factor, $k$-fold      284
Factor, double      284
Factor, invariant      361
Factor, multiple      284 287
Factor, multiple, isolation of      288
Factor, simple      284
Factor, single      284
Factor, triple      284
Factorization of polynomials      284
Factorization of polynomials into irreducible factors      281
Faddeyev, D.K.      414
Faddeyeva, V.N.      414
False position, method of      251
Ferrari, L.      12 230
Ferro, S. del      42
Fibonacci, L.      12; see Leonardo of Pisa
Field(s)      9 267
Field, characteristic of      270
Field, commutative      302
Field, concept of      257
Field, definition of      267
Field, division in, uniqueness of      267
Field, finite      268
Field, general theory of      13
Field, number      257 259 271
Field, obtained by adjoining an element      272
Field, of algebraic functions, theory of      10 13
Field, of algebraic numbers, theory of      10 13
Field, of complex numbers, construction of      273 275 295
Field, of complex numbers, uniqueness of      272
Field, of rational fractions      297ff
Field, of rational numbers      260 341
Field, splitting      296
Finite abelian groups      406ff
Finite Abelian groups, fundamental theorem on      406
Finite characteristic      271
Finite cyclic group      391
Finite fields      268
Finite group      383
Finite rings      268
Finite-dimensional linear space      183
Finite-dimensional spaces      182
Finite-dimensional unitary spaces      210
Form, cubic      306
Form, Jordan normal      370
Form, Jordan normal, reduction of a matrix to      375
Form, linear      62 306
Form, negative definite      177
Form, normal      169 170
Form, normal, of a matrix      355ff
Form, of degree $s$      306
Form, pairs of      219 223
Form, positive definite      174ff
Form, quadratic      306 (see also Quadratic form)
Form, quartic      306
Form, theory of      8
Form, trigonometric (of complex number)      114
Formula, Cardan’s      227 229
Formula, de Moivre’s      120
Formula, for differentiating a sum and a product      142
Formula, Lagrange interpolation      153
Formula, Newton’s      323
Formula, Taylor’s      145
Formula, Vieta’s      154 296 313
FOURIER      see Budan — Fourier Theorem
Fraction, partial      157
Fraction, rational      156 298
Fraction, rational, field of      297ff
Fraction, rational, in lowest terms      156
Fraction, rational, proper      156
Fraction, rational, simplified      156
Fraction, rational, symmetric      321
Fraction, symmetric rational      321
Fractional rational functions      156
Frazer, R.A.      414
Free unknowns      79
Frobenius, F.G.      13
Function, continuous      144
Function, fractional rational      156
Function, rational integral      337
Function, symmetric      312
Functional analysis      9 10
Fundamental system of solutions      84
Fundamental theorem of higher algebra      143
Fundamental theorem of the algebra of complex numbers      143ff
Fundamental theorem of the algebra of complex numbers, alternative proof of      337
Fundamental theorem on finite Abelian groups      406
Fundamental theorem on symmetric polynomials      314 316 319
Fundamental theorem on the similarity of matrices      367
Fundamental theorem, corollaries to      151ff
Galois, E.      9 12 13 232
Galois, theory of      11 13
Gantmacher, F.R.      414
Gauss, C.F.      12 143
Gauss’ (or Gaussian) elimination process      21
Gauss’ (or Gaussian) lemma      307 342
Gauss’ (or Gaussian) method      17 18 20
Gelfand, I.M.      414
Generate (verb) (subgroup generated by subgroups)      401
Generation of a linear subspace      196
Generator      390
Geometry, algebraic      9 326
Geometry, projective      11
Graeffe method      256
Grassmann, H.      13
Grave, D.A.      13 415
Greatest common divisor      131 133
Greatest common divisor of integers      133
Greatest common divisor of polynomials      134 135 138
Group(s)      10 382ff
Group(s), Abelian      383 385
Group(s), Abelian, finite      406ff
Group(s), Abelian, indecomposable      405
Group(s), Abelian, primary      407
Group(s), addition in      385
Group(s), additive      385
Group(s), alternating      387
Group(s), commutative      383
Group(s), continuous      11 13
Group(s), continuous, theory of      13
Group(s), cyclic      390ff
Group(s), cyclic, primary      407
Group(s), decomposition of      391ff
Group(s), definition of      382 383
Group(s), factor      394 395 396
Group(s), factor, examples      397
Group(s), finite      383
Group(s), finite Abelian      406ff
Group(s), finite Abelian, complete survey of      413
Group(s), finite Abelian, fundamental theorem on      406
Group(s), finite cyclic      391
Group(s), finite cyclic, general theory of      13
Group(s), finite cyclic, infinite cyclic      390 391
Group(s), finite cyclic, isomorphic      385
Group(s), finite cyclic, Lie      11
Group(s), finite cyclic, multiplication in      382
Group(s), finite cyclic, multiplicative      386 391
Group(s), finite cyclic, noncommutative      387
Group(s), finite cyclic, order of      383
Group(s), finite cyclic, primary      407
Group(s), finite cyclic, primary Abelian      407
Group(s), finite cyclic, primary cyclic      405 407
Group(s), finite cyclic, theory of      10 382
Group(s), finite cyclic, theory of, Soviet school of      14
Gurevich, G.B.      414
H$\ddot{o}$lder, O.      13
Hamilton      see Cayley-Hamilton theorem
Hamilton, W.R.      13
Hecke, E.      415
Height of a polynomial      353
Higher algebra      7 8
Higher-degree equations      231
Highest term of a polynomial      311
His$\hat{a}$b al-jabr w’al-mug$\hat{a}$-balah      12
Hodge, W. V.D.      415
Homogeneous linear equations      see Systems of homogeneous linear equations
Homogeneous polynomial      306
Homological algebra      13
Homomorphic mapping      397
Homomorphism(s)      394 397
Homomorphism(s), canonical      398
Homomorphism(s), theorem on      398
Horner method      140 141
Hurwitz, A.      13
Hypercomplex numbers, theory of      10
Hypercomplex systems, theory of      13
Ideals, theory of      10 13
Identity element of a group      383 385
Identity matrix      93
Identity permutation      31
Identity transformation      189 195 214
Image      188
Imaginaries, axis of      112
Imaginaries, pure      112
Imaginary part      112
Imaginary unit      112
Incomplete cubic equation      226
Inconsistent system of linear equations      16
IndecomposabiHty of groups      405
Indecomposable Abelian group      405
Indefinite quadratic forms      177
Indeterminate system      16
Index of a subgroup      393
Index of inertia, negative      172
Index of inertia, positive      172
Inertia, law of      169 170
Inertia, negative index of      172
Inertia, positive index of      172
Infinite cyclic group      390 391
Infinite-dimensional linear spaces      181
Infinite-dimensional spaces      9
Integers, system of      107
Integral rational functions      156
Interpolation formula, Lagrange      153
Interpolation, linear, method of      251
Invariant (adj.)      211
Invariant factors      361
Invariant subgroup      394
Invariants, theory of      9
Inverse (to a class)      301
Inverse element      179 384
Inverse linear transformation      199
Inverse matrices      93
Inverse matrix, left      94
Inverse matrix, right      94
Inverse of a permutation      33
Inverse operation      261
Inverse polynomial      129
Inverse transformation      199
inversion      29
Irrational numbers      107
Irreducible (of a polynomial)      281 306
Irreducible (of a solution)      230
Isomorphic (adj.)      272
Isomorphic correspondence      182
Isomorphic Euclidean spaces      208
Isomorphic groups      385
Isomorphic real linear spaces      181
Isomorphism(s)      178 181
Isomorphism(s) of Euclidean spaces      208
Isomorphism(s) of fields      272ff
Isomorphism(s) of rings      272ff
Iterative procedures      58
Jacobson, N.      414 415
Jordan matrices      370
Jordan matrix of order $n$      370
Jordan normal form      370
Jordan normal form, reduction of a matrix to      375
Jordan submatrix      371
Jordan, M.E.C.      13
Kernel of a homomorphism      398
Kernel of a linear transformation      197
Khayyam, Omar      12
Kronecker — Capelli theorem      77 78 81
Kronecker, L.      13 345
Kummer, E.E.      13
Kurosh, A.G.      415
Lagrange interpolation formula      153
Lagrange, J.L.      12 13
Lagrange’s Theorem      393
Laplace, P. S.      12
Laplace’s theorem      50 51
Lattice      11
Lattice theory      11 13
Law of inertia      169 170
Leading coefficient      126
Left coset      392
Left decomposition      392
Left inverse matrix      94
Left-identity      384
Left-inverse      385
Lemma      see Theorem
Lemma, d’Alembert’s      147 149
Lemma, Gauss’ (or Gaussian)      307 342
Lemma, on the increase of the modulus of a polynomial      146
Lemma, on the modulus of the highest-degree term      145
Leonardo of Pisa      12; see Fibonacci
Lie groups, theory of      11
Lie, S.      13
Linear algebra      7 8 13 15 276
Linear combination of vectors      62
Linear dependence of vectors      62ff
Linear equations      see Systems of linear equations
Linear form      62 306
Linear interpolation, method of      251
Linear polynomials      127 139
Linear spaces      7 178ff
Linear spaces, $n$-dimensional      185
Linear spaces, complex      202 209
Linear spaces, finite-dimensional      183
Linear spaces, infinite-dimensional      181
Linear subspaces      195ff 202
Linear subspaces, generation of      196
Linear substitution      87
Linear transformation(s)      87 89 188
Linear transformation(s), inverse      199
Linear transformation(s), kernel of      197
Linear transformation(s), nonsingular      93 198
Linear transformation(s), nonsingularity of      224
Linear transformation(s), null space of      197
Linear transformation(s), operations on      193
Linear transformation(s), product of      193
Linear transformation(s), product of, by a scalar      193
Linear transformation(s), rank of      197
Linear transformation(s), singular      93
Linear transformation(s), spectrum of      200
Linear transformation(s), with a simple spectrum      202
Linearly dependent system of vectors      63 64
Linearly independent system of vectors      63
Lobachevsky, method of      256
Lobachevsky, N.I.      13
Lyapin, E.S.      414 415
Maltsev, A.I.      414
Mapping, homomorphic      397
Matrices      see also Matrix
Matrices, $\lambda$-matrices      355
Matrices, $\lambda$-matrices, canonical      356 357
Matrices, $\lambda$-matrices, equivalence      355ff
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