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Swallow J. — Exploratory Galois Theory
Swallow J. — Exploratory Galois Theory



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Название: Exploratory Galois Theory

Автор: Swallow J.

Аннотация:

Swallow (mathematics, Davidson College) works from the assumption his readers are undergraduates who have completed a first course in abstract algebra in this exploration-based approach to Galois theory. In his text he covers algebraic numbers, field extensions, minimal polynomials, multiply generated fields, and the Galois correspondence, closing with such classical topics as binomial equations and solvability in radicals. He also gives students with access to Maple or Mathematica the opportunity to work with finite extensions of the rational numbers. Swallow includes exercises and an historical note.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 220

Добавлена в каталог: 23.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Action, group      17
Adjoined      25
Affine translation      44
Algebra, fundamental theorem of      12
Algebraic closure      179
Algebraic element      85 179
Algebraic extension      85 179
Algebraic number      22
Algebraic number, degree of      23
Algebraic number, representation of by an arithmetic combination      25
Algorithm, division      7
Algorithm, Division, for integral domains      20
Algorithm, Euclidean      9
Arithmetic combination      25
Associated polynomial      22
Associates      10
Automorphism, field      16
Automorphism, Frobenius      190
Automorphisms, independence of      157
Characteristic of a field      19
Circle, constructible      165
Closure, algebraic      179
Coefficient, leading      6
Combination, arithmetic      25
Combination, K[X]-linear      9
Conjugacy class of subgroups      130
Conjugate element      105
Conjugate number      42
Conjugate subgroup      130
Constant      6
Constant polynomial      6
Constructible circle      165
Constructible number      166
Constructible point      165
Criterion, Eisenstein irreducibility      21
Criterion, mod p irreducibility      20
Cyclotomic extension      153
Cyclotomic polynomial      154
Dedekind's Lemma      160
Degree of a polynomial      6
Degree of an algebraic number      23
Derivative      50
Discriminant      115 125 139
Discriminant resolvent      125
Division algorithm      7
Division Algorithm for integral domains      20
Division, polynomial      7
Divisor, greatest common      9 11
Domain, integral      10
Domain, principal ideal      11
Domain, unique factorization      10
Eisenstein irreducibility criterion      21
Element, algebraic      85 179
Element, conjugate      105
Element, primitive      64 84
Element, separable      181
Elementary symmetric polynomial      121
Euclidean algorithm      9
Evaluation      25
Evaluation homomorphism      32 68 69
Exponent of a group      174 189
Extension of isomorphism      78
Extension, algebraic      85 179
Extension, cyclotomic      153
Extension, field      26
Extension, finite      85
Extension, Galois      104 184
Extension, generated field      25 84
Extension, normal      103 182
Extension, radical      171
Extension, separable      181
Extension, simple      64 84
Extension, splitting      66 71 181
Extension, transcendental      85
factor [Maple]      13
Factor [Mathematica]      13
Factor, linear      12
Factor, polynomial      6
FDeclareExtensionField      53 91
FDeclareField      33 52 90
FDeclareSplittingExtensionField      92
FDeclareSplittingField      92
FFactor      55 94
FFindFactor      58
FFindFactorRt      58 96
FGaloisGroup      137
FGaloisResolvent      141
Field automorphism      16
Field extension      26
Field extension, algebraic      85 179
Field extension, generated      25 84
Field extension, normal      103 182
Field extension, separable      181
Field extension, simple      64 84
Field extension, splitting      66 71 181
Field generated by an algebraic number      25 43
Field generated by several algebraic numbers      63
Field monomorphism      16
Field of rational functions      85
Field operations      25
Field, characteristic of      19
Field, fixed      107
Field, Galois      189
Field, intermediate      108
Field, multiply generated      63
Field, number      86
Field, perfect      185
Field, splitting      66 71 181
Finite extension      85
Finitely generated field extension      84
FInvert      33 54 93
First Isomorphism Theorem for rings      15
Fixed field      107
FMakeTower      59 97
FMap      100
FMapIsIsoQ      99
FMinPoly      57 96
Form, reduced      26 30 87
FPolynomialExtendedGCD      57 95
FPolynomialGCD      57 95
FPolynomialOrbit      141
FPolynomialQuotient      56 94
FPolynomialRemainder      56 94
Frobenius automorphism      190
Frobenius map      185
FRootNumber      57 96
FSimplifyE      33 54 93
FSimplifyP      55 94
fsolve [Maple]      12
FSubstitutelnGaloisResolvent      143
Fundamental theorem of algebra      12
Galois extension      104 184
Galois field      189
Galois group      105
Galois resolvent      117
GCD      9
gcd [Maple]      14
gcdex [Maple]      14
Generated field      25
Generated field extension      84
Generated group      17
Generated ideal      24 67
Generation by an algebraic number      43
Greatest common divisor      9
Group action      17
Group, exponent of      174 189
Group, Galois      105
Group, generated      17
Group, permutation      17
Group, solvable      174
Group, symmetric      17
Homomorphism, evaluation      32 68 69
Homomorphism, trivial      16
Ideal      11
Ideal, generated      24 67
Identities, Newton's      121
Independence of automorphisms      157
Integral domain      10
Integral domains, Division Algorithm for      20
Intermediate field      108
Inverse Galois problem      106
Irrationalities, Natural, Theorem on      128
Irreducibility criterion, Eisenstein      21
Irreducible element      10
Irreducible polynomial      6
Isomorphism, extension of      78
Kronecker's theorem      178
Law, Tower      50
Leading coefficient      6
Least common multiple      20
Lemma, Dedekind's      160
Linear factor      12
Linear polynomial      12
Map, Frobenius      185
Minimal polynomial $m_{\alpha}$ of $\alpha$      23
Mod p irreducibility criterion      20
Monic      6
Monomial occurrence      122
Monomorphism, field      16
Multiple, least common      20
Multiplicity of a root      40
Multivariate polynomial      67
N [Mathematica]      13
Natural Irrationalities, Theorem on      128
Newton's identities      121
Normal extension      103 182
NSolve [Mathematica]      12
Number field      86
Number of operations      36
Number, algebraic      22
Number, algebraic, representation of by an arithmetic combination      25
Number, conjugate      42
Number, constructible      166
Number, root      89
Number, transcendental      23 86
Occurrence, monomial      122
Operations, field      25
Orbit      17
Orbit of a polynomial      117
Over      6 23 25 26 42 63 71 78 105
Perfect field      185
Permutation      17
Permutation group      17
Permutation representation      17
Point, constructible      165
Polynomial      5
Polynomial division      7
Polynomial factor      6
Polynomial orbit      117
Polynomial Quotient      7 20
Polynomial remainder      7
Polynomial ring      5
Polynomial ring in //variables      67
Polynomial solvable by radicals      172
Polynomial stabilizer      117
Polynomial, associated      22
Polynomial, constant      6
Polynomial, cyclotomic      154
Polynomial, elementary symmetric      121
Polynomial, irreducible      6
Polynomial, linear      12
Polynomial, minimal      23
Polynomial, monic      6
Polynomial, multivariate      67
Polynomial, quotient      20
Polynomial, reducible      6
Polynomial, remainder      20
Polynomial, separable      180
Polynomial, split      72
Polynomial, symmetric      120
PolynomialExtendedGCD [Mathematica]      15
PolynomialGCD [Mathematica]      14
PolynomialQuotient [Mathematica]      14
PolynomialRemainder [Mathematica]      14
Prime subfield      19 87
Primitive element      64 84
Primitive Element, Theorem of the      186
Primitive root of unity      152
Principal ideal domain      11
Problem, inverse Galois      106
Problem, subfield immersion      59 97
Product, semidirect      161
quo [Maple]      14
Quotient polynomial      7 20
Quotient, polynomial      7
Radical extension      171
Rational functions, field of      85
Rational root theorem      20
Reduced form      26 30 87
Reducible polynomial      6
Reduction modulo n\,^      28
rem [Maple]      14
Remainder polynomial      7
Remainder, polynomial      7
Representation of a number by an arithmetic combination      25
Representation, permutation      17
Resolvent, discriminant      125
Resolvent, Galois      117
Ring generated by an algebraic number      26
Ring, polynomial, in n variables      67
Ring, polynomial, in one variable      5
Root number      89
Root of unity      152
Root [Mathematica]      13 89
Root, multiplicity of      40
Rootof [Maple]      13 89
Semidirect product      161
Separable element      181
Separable extension      181
Separable polynomial      180
Simple extension      64 84
Solvable by radicals, polynomial      172
Solvable group      174
Split polynomial      72
Splitting field      66 71 181
Stabilizer      17
Stabilizer of a polynomial      117
Subfield immersion problem      59 97
Subfield, prime      19 87
Subgroup, conjugate      130
Subgroup, transitive      135
Symmetric group      17
Symmetric polynomial      120
Theorem of the Primitive Element      186
Theorem on Natural Irrationalities      128
Theorem, First Isomorphism, for rings      15
Theorem, Fundamental, of Algebra      12
Theorem, Kronecker's      178
Theorem, Rational Root      20
Tower Law      50
Transcendental      23
Transcendental extension      85
Transcendental number      86
Transitive subgroup      135
Translation, affine      44
Trivial homomorphism      16
Unique factorization domain      10
Unity, primitive root of      152
Unity, root of      152
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