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Nagata M. — Field Theory
Nagata M. — Field Theory

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Название: Field Theory

Автор: Nagata M.

Аннотация:

The theory of fields has traditionally been considered a basic part of abstract algebra. In modern mathematics, however, the abundance of algebraic ideas which have been introduced has established the importance of the theory of fields not only in algebra but throughout all areas of mathematics as well. There are many topics that require discussion in the mathematics courses frequently offered in colleges and universities. Consequently, it is not unusual to have too few lecture hours devoted to the theory of fields.
In view of this, the author wished to publish a book on field theory, rich in topical variety, necessitating few prerequisites, and conveniently sized, that would allow any student to advance his study.
In this book, it is assumed that the reader is familiar with the basic definitions and results on set theory and determinants, and therefore these results are stated explicitly and without proof. In addition, some basic results on group theory and ring theory will
be needed; proofs foij these results are given. Thus the main text of this volume initially consists of preliminaries on groups and rings (Chapters 1 and 2) and in subsequent chapters (Chapters 3 to 7) focuses on several important topics on
fields, such as algebraic and transcendental extensions, valuations, ordered fields, and Galois theory of algebraic extensions.


Язык: en

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1977

Количество страниц: 276

Добавлена в каталог: 16.05.2011

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$C_{i}$-field      134
$C_{i}'$-field      134
$T_{i}$-space      151
-adic      146 170
-adic expansion      150
-base      109
-independent      109
-th root      62
Abelian      9
Abelian extension      73
act      15 69
Addition      8
Additive group      9
Additive valuation      166
algebraic      55
Algebraic closure      55 88
Algebraic integer      88
Algebraic number      88
Algebraically closed      55 88
Algebraically dependent      33
Algebraically independent      33 91 98
Alternating      45
Alternating group      48
Approximation theorem      172 174
Archimedean      146 232
Artin module      44
Artin ring      44
Ascending central series      15
Ascending chain condition      5
Associated prime ideal      119
Associative ring      23
Associativity      8
Automorphism      12 26
Automorphism group      12
Basis      200
Belong      118
Bilinear      101
Binary operation      8
Canonical mapping      101 102
Cardinality      3
Cardinality of continuum      3
Cauchy sequence      160
Center      14
centralizer      14
Characteristic      24
Chinese remainder theorem      28
class      6
Classification      6
Closed set      150
Closed subgroup      235
Closure      154
Coefficient      32
Common divisor      35
Common factor      35
Common multiple      35
Commutative      8 9 23
Commutator      14
Commutator group      14
Commutator subgroup      14
Compact      155
complete      160
Complete set of conjugate      94
Complete valuation ring      186
Completion      160
Complex number field      24
Composite      171
composition      2
Composition series      42
Conductor      143
Congruent      16
conjugate      12 60
Containment relation      4
continuous      153 156
Converge      160
Coset      10
Countable      3
Countably infinite      3
Cyclic      9
Cyclic extension      73
Cyclic permutation      47
Cyclotomic field      73
Cyclotomic polynomial      75
Degree      11 32 55 91 142 179
Degree in appearance      32
Degree of inseparability      66
Degree of separability      66
Derivation      105 140
Derivative      61
Derived group      14
Derived normal ring      50
Derived series      14
Descending central series      14
Descending chain condition      5
Difference product      46 47
DIMENSION      43 98
Direct product      18 21 28 201
Direct sum      18 21 22 27 28
Discrete      155 181
Discriminant      81
Distance      153
Distance function      152
Divide      34 35
Divisible      34 35 205
Division ring      24
Divisor      35
Dual      4 201
Elementary symmetric form      46
Elementary symmetric polynomial      46
Endomorphism      26
Equivalence relation      6
Equivalent      160 167 171
Euclid algorithm      38
Euclid ring      36
Euler function      75
Even permutation      48
Expressible by radicals      78
Extension      54 160 176
Extension by radicals      78
Extension field      54
Factor      34 35
Factor Theorem      34
Field      24
Field of fractions      31
Field of quotients      31
finite      10 41 56
Finite field      56
Finite intersection property      155
Finite set      3
Finitely generated      41 99
FORM      134
Formal power series      202
Formally really      208
free      41 104
Free base      41
Frobenius map      203
Full matric algebra      25
Fundamental system of neighborhoods of the identity      157
Fundamental theorem of Galois      71 237
Galois extension      67 125
Galois group      68 125
Gauss plane      84
GENERATED      9 27 41 54
Generator      9
Geometric construction      84
Going-down Theorem      126
Going-up Theorem      124
Graded      142
greater      4
GREATEST      5
Greatest common measure      35
Group      9 15
Hausdorff      152
Height      124
Hensel lemma      186 187 195
Hensel ring      188
Henselization      190
Hilbert zero-point theorem      130
Homeomorphic      154
Homeomorphism      154
Homogeneous      142
Homogeneous form      32
Homomorphism      12 25 40
Ideal      26
Ideal quotient      118
Idempotent      51
Identity      8 24
Image      2
Independence theorem of additive valuations      174
INDEX      11
Induced      152
Induction      7
Inductive set      6
Inertia field      239
Inertia group      239
Inertia ring      239
Infimum      5
Infinite field      56
Injection      12 26 40 65 208
Injective      12 26
Inner automorphism      12
Inner automorphism group      12
Inseparable      61
integral      49
Integral closure      50
Integral domain      24
Integrally closed      50
Invariant      16 69
Inverse      2 9
Inverse image      2
Invertible      9 24
Irreducibility theorem of Eisenstein      53
Irreducible      35
Irredundant      119
Isolated subgroup      205
Isomorphic      12 26 208
Isomorphism      12 26 40 208
Isomorphism theorem      13 41
Jacobian matrix      107
Jacobson radical      181
Jordan — Hoelder — Schreier theorem      42
Kernel      12 25
Kronecker $\delta$      50
Krull dimension      127
Krull — Azumaya (lemma of)      181
Kummer (n-) extension      95
Lagrange resolvent      76
Larger      4
Largest      5
Largest common factor      35
Least common multiple      35
Left module      39
Left residue class      10
Left zero-divisor      24
Length      42 43 47
Lexicographical order      6
Lie over      123
Limiting point      160
linear combination      32
Linearly dependent      cf. "Linearly independent"
Linearly disjoint      102
Linearly independent      41
Linearly independent base      41
Linearly ordered      4
Local tensor product      102
Lower bound      5
Luroeth (theorem of)      137
Lying-over theorem      123
Mapping      2
Mathematical induction      7
Maximal      5
Maximal completion      226
Maximal ideal      30
Maximal ideal with respect to      30
Maximally complete      225
Maximum condition      5
Metric space      152
Minimal      5 120
Minimal polynomial      58
Minimal splitting field      59
Minimum condition      5
minus      9
Module      9 39 40
MODULO      10 16 26
Monic      32
Monomial      32
Multiple      35
Multiple root      39
Multiplication      8 39
Multiplicative valuation      145
Multiplicatively closed      29
Multiplicity      39
Nakayama (lemma of)      181
Natural      12 102 122
Natural topology      234
Neighborhood      150 152
Nilpotent      14 24 117
Noetherian      116
Nonarchimedean      145
Norm      94
Normal base      95
Normal extension      67 125
Normal ring      50
Normal space      152
Normal subgroup      12
Normalization theorem for finitely generated rings      129
Normalization theorem for polynomial rings      128
Normalization theorem of Noether      129
normalizer      20
Null-sequence      161
Odd permutation      48
Of finite type      41
Onto      2 26
Open neighborhood      150
Open set      150
Operation      8
Orbit      16
Order      4 10
Order of inseparability      112
Order-isomorphic      208
Order-isomorphism      208
Ordered additive group      166
Ordered field      208
Ordered group      166
Ordered set      4
Over      25 105
p-adic      146 170
p-base      109
p-group      15
p-independent      109
Partial derivation      106
Perfect      61
Permutation group      11
Place      171
Point      150
Polynomial      32
Polynomial ring      33 89
Power      27
Primary      117 118
Prime divisor      119
Prime element      35
1 2
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