Albert A.A. — Structure of algebras :: Электронная библиотека попечительского совета мехмата МГУ

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Albert A.A. — Structure of algebras

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Название: Structure of algebras

Автор: Albert A.A.

Аннотация:

The theory of linear associative algebras probably reached its zenith when the solution was found for the problem of determining all rational division algebras. Since that time it has been my hope that I might develop a reasonably self-contained exposition of that solution as well as of the theory of algebras upon which it depends and which contains the major portion of my own discoveries. The first step in carrying out this desire was necessarily that of writing a text with contents selected so as to provide a foundation adequate for the prospective exposition. This text has already been published under the title Modern Higher Algebra. Its completion was followed shortly by the timely invitation of the Colloquium Committee of the American Mathematical Society to me to write these Lectures embodying the desired exposition.

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Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1939

Количество страниц: 220

Добавлена в каталог: 16.05.2011

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Предметный указатель

$J_{\mathfrak{F}}$ -array      130
$\mathfrak{D}$ -basis      14
Absolute irreducibility      121
Algebraic number theory lemmas      147
Algebras      3
Algebras of order one      20
Algebras with a total matric subalgebra      19
Algebras, characteristic function of      11 16 122
Algebras, components of      28
Algebras, degree of      16
Algebras, equivalence of      3
Algebras, equivalence to matric algebras      11
Algebras, general quantity of      16
Algebras, principal degree of      16
Algebras, reciprocal      3
Applications of the Galois theory      63
Archimedean valuations      145
Automorphisms      7
Automorphisms of a direct product      85
Automorphisms of a field      35
Automorphisms of a normal simple algebra      51
Automorphisms of an unramified field      141
Automorphisms, extension of equivalences to      54
Automorphisms, inner      9 49
Automorphisms, reciprocal      8
Basis of integers      130
Basis of total matric algebra      7
Bibliography      193ff
Centrum      6
Centrum of a direct sum      30
Centrum of a simple algebra      41
Centrum of an involutorial algebra      153
Centrum, scalar extension of      35
Characteristic function      11 16 122
Class group      59
Classes of algebras      58
Commutative semi-simple algebras      39 40 44
Commutator algebra      6
Commutator of equivalent subalgebras      56
Completely ramified algebras      137
Components of algebras      28
Components of representations      115
COMPOSITES      34
Construction of crossed products      68 182
Construction of involutions      157
cosets      27
Crossed products      66
Crossed products of degree four      187
Crossed products, construction of      68 182
Crossed products, direct products of      71
Crossed products, equivalence of      71
Crossed products, involutorial      158
Crossed products, normalized      73
Crossed products, scalar extension of      73
Cyclic algebras      74
Cyclic algebras, direct factorization of      100
Cyclic algebras, direct powers of      75 97
Cyclic algebras, direct products of      74
Cyclic algebras, equivalence of      75
Cyclic algebras, exponents of      98
Cyclic algebras, total matric      74
Cyclic fields      35
Cyclic p-algebras      107
Cyclic semi-fields      83
Cyclic semi-fields, direct products of      86
Cyclic semi-fields, norms in      84
Cyclic systems      88
Cyclic systems, group of      89
Cyclic systems, powers of      91
Decomposable GR-matrices      126
Decomposable representations      115
Degree of a cyclic system      88
Degree of a normal simple algebra      43
Degree of a splitting field      60
Degree of an algebra      16
Diagonal algebras      20 44 79
Diagonal direct factors      82
Difference algebra      28
Difference group      27
Direct factorization of algebras      43
Direct factorization of cyclic algebras      100
Direct factorization of normal division algebras      77
Direct factorization, uniqueness of      85
Direct powers      6
Direct powers of cyclic algebras      75 97
Direct powers of division algebras      75
Direct products      5
Direct products of crossed products      71
Direct products of cyclic algebras      75
Direct products of normal algebras      41
Direct products of normal division algebras      52
Direct products of total matric algebras      7 50
Direct sum      28
Direct sum, centrum of      30
Direct sum, ideals of      29
Direct sum, semi-simple      39
Division algebras      14
Division algebras, commutative      15
Division algebras, direct product of      52
Division algebras, normal      41
Division algebras, ramified      137
Division algebras, subalgebras of      15
Division p-adic algebras      137
Enveloping algebra      113
Equivalence of algebra-group pairs      65
Equivalence of algebras      3
Equivalence of crossed products      71
Equivalence of cyclic algebras      75
Equivalence of representations      110
Exponent of a field      33
Exponent of normal simple algebra      76
Exponent of p-adic algebra      144
Exponent of p-algebra      109
Exponent of rational division algebra      150
Extension of equivalences      54
Factor sets      67
Fields, inseparable      32 101
G-irreducible algebra      78
General quantity      16
Generalized cyclic algebras      93
Generalized cyclic algebras, Riemann matrices      126
Generating automorphism      35
Generating automorphism of a cyclic semi-field      83
Group of classes      59
Group of cyclic systems      89
Group-algebra pairs      65
Grunwald lemma      148
Hensel lemma      136
Ideals      22
Ideals of a direct sum      29
Ideals of a nilpotent algebra      22
Ideals of a p-adic algebra      133
Ideals, maximal      38
Ideals, prime      134
Idempotents      20
Idempotents, existence of      23
Idempotents, primitive      26
Idempotents, principal      25
Idempotents, rank of      50
Idempotents, similar      50
Impure GR-matrices      126
Impure representation algebras      114
Impure Riemann matrices      188
Index of a class      58
Index of a nilpotent quantity      22
Index of a normal simple algebra      58
Index of a p-adic extension      148
Index of an algebra      4 5
Index reduction factor      59
Inseparable      32

Integers      129
Integral basis      130
Integral domains      129
Intersection of ideals      22
Intersection of left ideals      21
Intersection of linear sets      1
Invariant subspace      113
Involutions      151
Involutions of direct products      155
Involutions of normal simple algebras      156
Involutions of similar algebras      156
Involutions of simple algebras      154
Involutions, construction of      157
Involutions, kinds of      153
Involutorial algebras      151
Involutorial algebras, centrum of      153
Involutorial algebras, symmetric subfields of      157
Involutorial crossed products      158
Involutorial cyclic algebras      160
Involutorial quaternion algebras      161
Involutorial rational algebras      161 162
Isomorphic matrices      126 188
J-skew quantities      152
J-symmetric quantities      151
Least representation algebra      61
Left ideal      21
Left ideal, simple left ideal      23
Left linear set      14
Left multiplications      12
Linear sets      1
Linear sets, products of      3
Linear sets, sums of      1
Linear sets, supplementary      2
Linear sets, zero      1
Linear transformations      9 111
Matric basis      7
Matric representation      111
Maximal $J_{\mathfrak{F}}$ -array      130
Maximal ideal      38
Maximal subfields      56 85
Minimum function      11 13 16 122
Multiplication algebras      128 189
Multiplication algebras, structure of      168
Multiplication algebras, subfields of      166
Nilpotent algebra      22
Non-associative algebras      188
Non-singular quantity      13
Non-singular transformation      10
Norm condition      184
Norm function      18 122
Normal algebras      6
Normal algebras over centrum      15
Normal algebras, direct products of      41
Normal direct factors      41
Normal division algebras      41
Normal division algebras of degree $p^{2}$       179
Normal division algebras of degree four      179
Normal division algebras of degree three      177
Normal division algebras of low degrees      178
Normal division algebras of prime degree      177
Normal division algebras, direct factorization of      77
Normal division algebras, exponent of      77
Normal division algebras, maximal subfields of      57
Normal division algebras, powers of      75
Normal division algebras, primary      77 174
Normal simple algebras      41
Normal simple algebras as direct factors      51
Normal simple algebras of degree two      146
Normal simple algebras, automorphisms of      51
Normal simple algebras, commutator of subfields of      53
Normal simple algebras, degree of      43
Normal simple algebras, direct product with reciprocal of      47
Normal simple algebras, exponent of      76
Normal simple algebras, index of      58
Normal simple algebras, involutions of      156
Normal simple algebras, maximal subfields of      56
Normal simple algebras, metacyclic subfields of      63
Normal simple algebras, scalar extension of      43 56 59
Normal simple algebras, simple subalgebras of      53
Normal simple algebras, splitting fields of      60
Normal simple algebras, subfields of      53
Norms in cyclic semi-fields      84
Order of a p-adic integer      132
Order of a p-adic quantity      133
Order of a product of sets      4
Order of an algebra class      76
Orthogonal quantities      20
p-adic division algebras      132
P-adic fields      138 151
p-adic normal simple algebras      142
p-algebras      104
Pairwise orthogonal      20
Peirce decompositions      24
Powers of algebras      4 5 75 97
Powers of cyclic systems      91
Primary algebra      77 174
Prime ideal      134
Primitive idempotent      26
Principal degree      16
Principal function      16
Principal idempotent      25
Principal matrix      165 188
Principal norm      18 124
Principal theorem      47
Principal trace      18 124
Products of classes      58
Products of cyclic systems      89
Products of factor sets      68
Products of ideals      22
Products of linear sets      3
Properly nilpotent quantity      24
Pure GR-matrix      126
Pure inseparable      32 101
Pure representation algebra      114
Pure Riemann matrix      188
Quadrate algebra      17
Quadratic field      145
Quaternion algebras      145
Radical      23
Radical of $e\mathfrak{U}e$       25
Radical of a direct sum      29
Ramification order      135
Ramification order of a division algebra      136
Ramification order of a prime ideal      148
Rank of idempotents      50
Rational division algebras      129
Rational division algebras, determination of      149
Rational division algebras, equivalence of      150
Rational division algebras, exponent of      150
Rational involutorial algebras      161
Real quaternion algebras      146
Real Riemann matrices      190
Reciprocal algebras      3
Reduced degree      33
Reduced field      173
Reduced norm      122 124
Reduced trace      122 124
Reducible algebras      28
Reducible representations      113
Reducible representations, components of      116
Regular quantity      13
Regular representation      11
Representation of algebras      110
Representation of field by algebra      60
Representations      111
Representations, absolutely irreducible      121
Representations, decomposable      115
Representations, fully decomposable      118
Representations, reducible      113
Residue-class degree      135

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