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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
Abel, N. H.      12 232
Abelian group      383 385
Abelian group, finite      406ff
Abelian group, finite, complete survey of      413
Abelian group, finite, fundamental theorem on      406
Abelian group, indecomposable      405
Abelian group, primary      407
Absolute value      113
Addition      261
Addition in a group      385
Addition matrix      99
Addition of classes      268
Additive groups      385
Additive notation      400
Adjoint of a matrix      94
Affine space      178
Affine transformations      214
al — Khowarizmi, Muhammad      12
Aleksandrov, P. S.      414
Algebra of complex number, fundamental Algebra, theorem of      143
Algebra of polynomials over an arbitrary field      276
Algebra, differential      11
Algebra, elementary      7
Algebra, higher      7 9
Algebra, homological      13
Algebra, linear      7 8 13 15 276
Algebra, of matrices      87ff
Algebra, of polynomials      8 9 13
Algebra, subject of      9
Algebra, tensor      9
Algebra, theorem of, alternative proof of      337ff
Algebra, theory of      13
Algebra, topological      11
Algebra, universal      11
Algebra, universal, theory of      13
Algebraic geometry      9 326
Algebraic numbers      349ff
Algebraic numbers, conjugate      350
Algebraic numbers, set of      350
Algebraic operation      261
Algebraic over a field (e.g. element)      279 305
Algebraic structures      9
Algebraic study, subject of      9
Algebraic systems      10
Algebraic systems, ordered, theory of      11
Algebraically dependent (of a system of elements)      305
Algebraically independent (of a system of elements)      305
Algebraization of mathematics      11
Algorithm, division      129 131
Algorithm, Euclid’s      133 136 241
Algorithm, of successive division      133
Alphabetical method      310
Alphabetical order of terms of a polynomial      310 311
Alternating group      387
Analysis, functional      9 10
Angle, polar      113
Angle, vectorial      113
Annihilate (e.g. polynomial annihilated by a linear transformation)      381
Approximate calculations, theory of      58
Approximation of roots      250ff
Argument of a complex number      114
Argument of a product of complex numbers      115
Argument of a quotient of complex numbers      116
Arrangement(s)      27ff; see Permutations
Arrangement, definition of      28
Arrangement, even      29
Arrangement, odd      29
Aryabhata      12
Associativity of addition      108
Associativity of matrix multiplication      96
Associativity of multiplication      109
Augmented matrix      21
Axiomatic definition      103
Axis of imaginaries      112
Axis of reals      112
B$\hat{o}$cher. M.      414
Baer, Reinhold      414
Bases      182
Bases, orthogonal      207
Bases, orthonormal      204 208
Bases, relationships between      185
Basis      see Bases
Bhascara      12
Binomial coeificient      121
Binomial theorem, Newton’s      120
Birkhoff, Garret      414
Brahmagupta      12
Budan — Fourier theorem      246 249
Canonical $\lambda$-matrix      356 357
Canonical homomorphism      398
Canonical quadratic form      164
Capelli      see Kronecker-Capelli theorem
Cardan, J.      12
Cardan’s formula      227 229
Cartan, Henri      414
Categories, theory of      13
Cauchy, A. L.      13
Cayley numbers      111
Cayley — Hamilton theorem      380 381
Cayley, A.      13
Change-of-basis matrix      185 186
Characteristic determinants      78
Characteristic matrix      199
Characteristic of a field      270
Characteristic polynomial of a matrix      200
Characteristic roots      199ff 216
Characteristic zero      270
Characteristic, finite      271
Chebotarev, N.G.      14 414 415
Chevalley, Claude      415
Ching Chou-chan      12
Ch’ang Ts’ang      12
Ch’in Chiu-Shao      12
Circle, closed      150
Class(es), addition of      268
Class(es), multiplication of      268
Class(es), opposite      294
Class(es), product of      293 301
Class(es), sum of      293 300
Class(es), unit      294 301
Class(es), zero      294 300
Closed circle      150
Coefficient, binomial      121
Coefficient, leading      126
Cofactors      43ff
Collar, A.R.      414
Common divisor      133
Commutative field      302
Commutative group      383
Commutative ring      267
Commutativity of addition      108
Commutativity of multiplication      109
Complementary minors      43
Complex linear spaces      181 202 209
complex numbers      107 110 112ff
Complex numbers, raising to a power      120
Complex numbers, taking roots of      120 122 123
Complex numbers, taking the square root of      122
Complex plane      112
Component(s) of an element      400
Component(s), primary      407
Congruent modulo $n$      268
Conjugate (the conjugate of $\alpha$)      118
Conjugate complex numbers      118
Conjugate elements      395
Conjugate numbers      119
Consistent system of linear equations      16
Constructive definition      103
Continuous function      144
Continuous groups, theory of      11 13
Correspondence, isomorphic      181
Coset, left      392
Countable set      352
Cramer, G.      12
Cramer’s Rule      24 53ff 56 57
Cramer’s rule, new derivation of      97
Criterion, Eisenstein      344 345
Cubic equations      226
Cubic equations, incomplete      226
Cubic equations, with real coefficients      228
Cubic form      306
Cubic polynomial      127
Cycle length      35
Cycle(s)      34
Cycle(s) of degree $n$      35
Cycle(s) of degree $n$, disjoint      35
Cyclic groups      390ff
Cyclic groups, finite      391
Cyclic groups, infinite      390 391
Cyclic groups, primary      405 407
Cyclic permutation      34
Cyclic subgroups      389
Cyclotomic polynomial      345
De Moivre’s Formula      120
Decomposability of a finite Abelian group      407
Decomposable quadratic forms      172
Decomposition      34
Decomposition, direct      400
Decomposition, into cycles      34
Decomposition, left      392
Decomposition, of a group      391ff 407
Decomposition, of polynomials      284
Decomposition, right      392
Decomposition, summands of      400
Decomposition, unique (of a proper rational fraction)      159
Decomposition, unique (of a proper rational fraction)example of      160
Decrement of a permutation      26
Dedekind, R.      13
Definiteness of a form, positive      177
Definition, axiomatic      103
Definition, constructive      103
Degree of a $\lambda$-matrix      365
Degree of a polynomial      303
Degree of a term      303
Denumerable set      352
Dependence of vectors, linear      62ff
Derivative of a polynomial      141
Derivative of a polynomial, second      141
Descartes, R.      12
Descartes’ Rule of Signs      247
Descartes’ theorem      247 249
Determinant(s)      23
Determinant, axiomatic construction of      103
Determinant, characteristic      78
Determinant, definition of      23
Determinant, evaluating      462
Determinant, expansion of      47
Determinant, multiplication theorem for      93
Determinant, of $n$th order      36ff
Determinant, of a system      54
Determinant, of second and third order      22ff
Determinant, second-order      23
Determinant, skew-symmetric      42
Determinant, theory of      12
Determinant, third-order      25 37
Determinant, Vandermonde      49 329 336
Determinate system      16
Diagonal matrices      371
Diagonal, principal      16
Diagonalization of a matrix      203
Difference      261
Differential algebra      11
Differentiating a sum and a product, formulas for      142
Dimension of a space      185
Diophantos of Alexandria      12
Direct decomposition      400
Direct product      406
Direct sum      400 403
Discriminant      326 334
Discriminant of a quadratic equation      335
Discriminant of an equation      228
Disjoint cycles      35
Dividend (of a polynomial)      306
Divisibility of polynomials      131 —133 305
Divisible      131
Divisible, exactly      131
Division algorithm      129 131
Division in a field, uniqueness of      267
Divisor(s)      131
Divisor, common      133
Divisor, elementary (of a matrix)      376
Divisor, elementary (of a polynomial)      376
Divisor, greatest common      131 133
Divisor, greatest common, of integers      133
Divisor, greatest common, of polynomials      133 135 138
Divisor, normal      394 398
Divisor, of a polynomial      306
Divisor, of unity      285
Duncan, W.J.      414
d’Alembert, J.R.      12
d’Alembert’s lemma      147 149
Eigenvalues      199
Eigenvector      200
Eilenberg, S.      414
Eisenstein criterion      344 —345
Element, component of      400
Element, conjugate      395
Element, identity (of a group)      385
Element, inverse      179 384
Element, multiples of      389
Element, of a set      261
Element, of infinite order      389
Element, opposite      179
Element, power of      389
Element, prime (of a ring)      285
Element, unit      269 383 385
Element, zero      180 264
Elementary algebra      7
Elementary divisors a polynomial      376
Elementary divisors of a matrix      376
Elementary matrix      363
Elementary symmetric polynomials      313
Elementary transformations      74
Elementary transformations of a matrix      355
Elimination of unknown      326 331
Equalizing coefficients, method of      23
Equation(s), $n$th-degree      232
Equation(s), cubic      226
Equation(s), cubic, incomplete      226
Equation(s), cubic, with real coefficients      228
Equation(s), general theory of      12
Equation(s), higher-degree      231
Equation(s), homogeneous linear      see Systems of homogeneous linear equations
Equation(s), nonhomogeneous      see System of nonhomogeneous equations
Equation(s), of second, third, fourth degree      225ff
Equation(s), quadratic      225
Equation(s), quartic      230
Equation(s), quintic      232
Equation(s), solvability of by radicals      12
Equation(s), systems of linear      see Systems of linear equations
Equivalence of $\lambda$-matrices      355ff
Equivalence relation      356
Euclidean algorithm      see Euclid’s algorithm
Euclidean space(s)      204
Euclidean space(s), $n$-dimensional      204 205
Euclidean space(s), isomorphic      208
Euclidean space(s), isomorphism of      208ff
Euclid’s algorithm      133 136 241
Expansion of a determinant      47
Extensions      271ff
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