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Howie J.M. — Fields and Galois Theory
Howie J.M. — Fields and Galois Theory

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Название: Fields and Galois Theory

Автор: Howie J.M.

Аннотация:

Fields are sets in which all four of the rational operations, memorably described by the mathematician Lewis Carroll as "perdition, distraction, uglification and derision", can be carried out. They are assuredly the most natural of algebraic objects, since most of mathematics takes place in one field or another, usually the rational field Q, or the real field M, or the complex field С This book sets out to exhibit the ways in which a systematic study of fields, while interesting in its own right, also throws light on several aspects of classical mathematics, notably on ancient geometrical problems such as "squaring the circle", and on the solution of polynomial equations.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2005

Количество страниц: 225

Добавлена в каталог: 09.02.2014

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$K$-automorphism      94
$p$-group      159
Abel, Niels Henrik (1802-1829)      3
Abelian basis theorem      151
Abelian group      3 149
Abel’s theorem      145
Addition (of polynomials)      34
Algebraic element      59
Algebraic extension      59
Algebraic independence      175
Algebraic number      59
Alternating group      162
Associates      3
Associative law      1
Automorphism      9
Automorphism, Frobenius      90
Automorphism, identity      94
Basis Theorem      151
Canonical extension (of an isomorphism)      37
Cantor, Georg Ferdinand Ludwig Philipp (1845-1918)      61
Cardano, Girolamo (1501-1576)      127
Centraliser      157
Centre (of a group)      157
Characteristic      17
Characteristic polynomial      64
Class equation      156
Commutative law      1 3
Commutative ring      2
Conjugate elements      156
Constructible point      75 183
Constructible polygon      189
Coprime      26 28
Coset      21
Coset, left      21
Coset, right      22
Cubic equation      128
CYCLE      160
Cyclic extension      140
Cyclic group      21
Cyclotomic polynomial      133 135
Dedekind, Julius Wilhelm Richard (1831-1916)      92
Degree of a polynomial      34
Degree of an extension      52
Degree, transcendence      176
Descartes, Ren$\acute{e}$ (1596-1650)      127
Dimension (of a vector space)      52
Direct product      153
Direct sum      149
Disjoint cycles      160
Division      4
Division algorithm      25
Divisor      4
Divisor, proper      4
Domain      2
Domain, Euclidean      25 33 38
Domain, factorial      30
Domain, integral      2
Domain, principal ideal      26
Domain, unique factorisation      30
Duplicating the cube      74 77
Eisenstein, Ferdinand Gotthold Max (1823-1952)      46
Eisenstein’s Criterion      46
Embedding      9
Equation, cubic      128
Equation, quadratic      127
Equation, quartic      130
Equation, quintic      173
Equivalence relation      5
Euclid of Alexandria (c. 325-265 B.C.)      25 187
Euclidean algorithm      27 39
Euclidean Domain      25 33 38
Extension      18 51
Extension by radicals      132
Extension, algebraic      59
Extension, cyclic      140
Extension, finite      52
Extension, finitely generated      175
Extension, Galois      115
Extension, infinite      52
Extension, normal      103
Extension, separable      110
Extension, simple      55
Extension, transcendental      59 175
Factor      4
Factor group      22
Factor, proper      4
Factorial domain      30 39
Fermat prime      191
Fermat, Pierre de (1601-1665)      191
Ferrari, Lodovico (1522-1565)      127
Ferro, Scipione del (1465-1526)      127
Field      2
Field, finite      85
Field, Galois      87
Field, of algebraic numbers      60
Field, of fractions      15
Field, perfect      110
Field, splitting      79 86
Finite extension      52
Finite field      85
Formal derivative      85
Frobenius automorphism      90
Frobenius, Ferdinand Georg (1849-1917)      90
Fundamental theorem of algebra      42
Galois correspondence      99
Galois extension      115
Galois field      87
Galois group      94
Galois, Evariste (1811-1832)      1
Gauss, Johann Carl Friedrich (1777-1855)      29 44 187
Gaussian integers      29 33
Gauss’s Lemma      44
General polynomial      178
Greatest common divisor      26 39
Group      3
Group, abelian      3 149
Group, alternating      162
Group, cyclic      21
Group, finite      21
Group, Galois      94
Group, of automorphisms      94
Group, of prime power order      157
Group, of units      3
Group, realisable      180
Group, simple      164
Group, soluble      154 163 167
Group, symmetric      160
Hermite, Charles (1822-1901)      61 180
Highest common factor      26
Hilbert, David (1862-1943)      140
Homomorphic image      22
Homomorphism      8
Homomorphism, natural      10 22
Homomorphism, substitution      40
Ideal      6
Ideal, principal      7
Identity      2
Identity automorphism      94
Indeterminate      35
INDEX      157
Infinite extension      52
Integral domain      2
Irreducible      29 39
Isomorphic      9
Isomorphism      9 22
Kernel      9 22 100
Klein, Felix Christian (1854-1912)      180
Lagrange, Joseph-Louis (1736-1813)      21 122
Lagrange’s Theorem      21
Leading coefficient      34
Lindemann, Carl Louis Ferdinand von (1852-1939)      61 78
Linearly independent      92
Liouville, Joseph (1809-1882)      61
Minimum polynomial      57
Monic polynomial      34
Monomorphism      9 92
Multiplication (of polynomials)      34
Natural homomorphism      10 22
Norm      140
Normal closure      106
Normal extension      103
Normal subgroup      22
Nullity      100
Partition      21
Perfect field      110
Permutation      162
Permutation, even      162
Permutation, odd      162
Perpendicular bisector      71
Poincar$\acute{E}$, Jules Henri (1854-1912)      180
Polynomial      33
Polynomial ring      35
Polynomial, characteristic      64
Polynomial, constant      34
Polynomial, cubic      34 128
Polynomial, cyclotomic      133 135
Polynomial, elementary symmetric      177
Polynomial, general      178
Polynomial, linear      34
Polynomial, minimum      57
Polynomial, monic      34
Polynomial, quadratic      34 127
Polynomial, quartic      34 130
Polynomial, quintic      34 173
Polynomial, separable      110
Polynomial, sextic      34
Polynomial, symmetric      177
Prime subfield      18 86
Principal ideal      7
Principal ideal domain      26 39
Quartic equation      130
Quintic equation      173
Quotient      25
Quotient group      22
Radical extension      132
Rank      100
Rational forms      36
Realisable group      180
Reflexive property      5
Regular polygon      187
Relatively prime      26
Remainder      25
Remainder theorem      40
Residue class      10
Residue class ring      10
Ring      1
Ring, commutative      2
Ring, with unity      2
Root (of a polynomial)      40
Ruler and compasses      71 75
Separable extension      110
Separable polynomial      110
Shafarevich, Igor Rostislavovich (1923-)      180
Simple extension      55
Simple group      164
Soluble by radicals      132 169 170 172
Soluble group      154 163 167 169 170 172
Solution by radicals      131
Splitting field      79 86
Splitting field, uniqueness      81
Squaring the circle      74 78
Subfield      6
Subfield, prime      18
Subgroup      21
Subgroup, normal      22
Subgroup, Sylow      156
Subring      6
Substitution homomorphism      40
Sylow subgroup      156
Symmetric group      160
Symmetric polynomial      177
Symmetric property      5
Tartaglia, Nicolo (1499-1557)      127
Trace      140
Transcendence degree      176
Transcendental element      175
Transcendental extension      59
Transcendental number      59
Transitive property      5
Transposition      161
Trisecting the angle      74 77
Unique factorisation domain      30 39
UNIT      3
Unity element      2
Vector space      51
Zero (of a polynomial)      40
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