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Hadlock C.R. — Field theory and its classical problems
Hadlock C.R. — Field theory and its classical problems

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Название: Field theory and its classical problems

Автор: Hadlock C.R.

Аннотация:

From author: "I wrote this book for myself.
I wanted to piece together carefully my own path through Galois Theory, a subject whose mathematical centrality and beauty I had often glimpsed, but one which I had never properly organized in my own mind. I wanted to start with simple, interesting questions and solve them as quickly and directly as possible. If related interesting questions arose along the way, I would deal with them too, but only if they seemed irresistible. I wanted to avoid generality for its own sake, and, as far as practicable, even generality that could only be appreciated in retrospect. Thus, I approached this project as an inquirer rather than as an expert, and I hope to share some of the sense of discovery and excitement I experienced. There is great mathematics here.
In particular, the book presents an exposition of those portions of classical field theory which are encountered in the solution of the famous geometric construction problems of antiquity and the problem of solving polynomial equations by radicals..."


Язык: en

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1978

Количество страниц: 336

Добавлена в каталог: 04.12.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\mathbf{C}$, automorphisms of      129 291
Abel, Niels Henrik      5 177 178
Abel, theorem of      165 172 181
Abelian group      138
Adamson, I.T.      179
Affine plane curve      186
Alexanderson, Gerald L.      120 179
Algebraic extension      72 79 80 258
Algebraic numbers      35
Algebraic numbers, countability      39 239
Algebraic numbers, form a field      36 46 80 241 259
Algebraically independent elements      141 211 216
Alternating function      243
Analytic, at a point      185
Analytic, at infinity      185
Analytic, at zero      184
Angles with rational degree measure, constructibility      119 275
Archibald, R.C.      120
Argument of a complex number      87
Artin, E.      127 177 178 179
Automorphisms      128
Automorphisms of $\mathbf{C}$      141 290 291
Automorphisms of $\mathbf{R}$      141 290
Automorphisms, associativity      130
Automorphisms, commutativity      129
Automorphisms, extensions of      141 290 295
Automorphisms, group of      131
Automorphisms, identity      129
Automorphisms, inverse      130
Basic Hilbert set      210 315
Bell, E.T.      7
Belonging to an exponent modulo p      102
Bieberbach, Ludwig      57
Birkhoff, Garrett      58 179
Bisection of an angle      30 232
Boyer, Carl      7
Brown, James W.      58 220
Cajori, Florian      178
Cancellation property, for polynomials      70
Cancellation property, modulo n      99
Cardano, Girolamo      178
Chain      290
Chain rule      56 247
Churchill, Ruel V.      58 220
Circle in a field $\mathbf{F}$      16
Clark, Allan      120 179
Closed      12
Coefficient field      126
Commutator      171
Complementary m-sets      108
Complementary periods      108
Completely reducible      62
Complex conjugate      80
Complex conjugate roots, of polynomials over $\mathbf{R}$      80 261
Complex conjugation, automorphism      129
Composition series      152
Congruences      98
Conjugacy, equivalence relation      147 150 294
Conjugacy, of elements      137
Conjugacy, of field extensions      147 294
Conjugacy, of subgroups      147 294
Constructibility      11 57 92
Constructibility and degree      81
Constructibility, compass alone      57
Constructibility, degree condition not sufficient      127 218 282
Constructibility, necessary and sufficient condition      21 94
Constructibility, necessary condition      95
Coset      133 135 284
Courant, Richard      57
Coxeter, H.S.M.      120
Cramer's rule      156
Cubic equation      126 176 177 279 304
Cubic equation, discriminant      177 305
Cubic equation, solvability by real radicals      177 307
Curiosity      24
CYCLE      166
Cyclic group      134
Cyclic group, generator      135
Cyclic group, subgroups      136 286
Cyclotomic polynomials      96 174 252
Cyclotomic polynomials, Galois group      140
Cyclotomic polynomials, irreducibility      119 120 277
Dedekind, R.      177 178
Degree, of a field extension      72
Degree, of a polynomial in one variable      34
Degree, of a polynomial in several variables      39 40
Degree, of a simple extension      74
Degree, of a succession of field extensions      78
Degree, of an algebraic number over a field      73
Dehn, Edgar      178
del Ferro, Scipione      5
Delian problem      3
Dense      210 315
Derivatives of complex functions      49 50 55
Diamond, Harold, G.      58
Dickson, L.E.      120 178
Discriminant of a cubic      177 305
Divisibility of polynomials      34
Division algorithm, for $\mathbf{Z}$      66 70 250
Division algorithm, for polynomials      60
Divisor      61
Doerge, Karl      220
Doubling the cube      2 24
Duarte, F.J.      4 58
e, transcendence of      56 248
Eichler, Martin      220
Eigenvalue      37
Eisenstein irreducibility criterion      64
Elementary symmetric functions      41
Elliptic functions      178
Equivalence relation      99 150
Euclidean algorithm, for $\mathbf{Z}$      66 70 250
Euclidean algorithm, for polynomials      67
Euler $\phi$-function      85 100
Exponential function, complex argument      47
Extension      58 71 72
Extension, algebraic      72 79 258
Extension, degree of      72
Extension, finite      71 75
Extension, multiple      71 72
Extension, normal      141 142
Extension, radical      124
Extension, simple      71 72
Extension, transcendental      72
Factorization into irreducible factors      69
Factorization into irreducible factors, algorithm for $\mathbf{Q}[x]$      71 255
Fermat primes      97
Fermat's theorem      100
Ferrari, Ludovico      5
Field      12 34 177
Field isomorphism      290
Finite extension      71 75
Finite fields      120
Finkbeiner, Daniel T., II      120
Fixed field, of a group      139
Fixed field, of an automorphism      139
fun      8
Fundamental constructions      9
Fundamental Galois pairing      145
Fundamental theorem of algebra      34 57
Fundamental theorem of calculus      50 247
Fundamental Theorem of Galois Theory      147 149
Fundamental Theorem on Symmetric Functions      41
g.c.d.      66
Gaal, Lisl      179
Galois group      140
Galois group, computation of      164 165 174 178 298
Galois pairing      145
Galois, Evariste      5 177 178
Galois, theorem of      151 152
Gauss, Carl Friedrich      3 57 107 119 120
Gauss, lemma of      62 197 310
Generator      134
Generic polynomial of degree n      216
Goldstein, Larry Joel      179
Greatest common divisor, of polynomials      66
Greek problems      2 9
Group      131
Group, of a field extension      136
Hasse, Helmut      179
Hermes      120
Hermite, C.      178
Herstein, I.N.      179
Hewitt, Edwin      58
Highest term      42
Hilbert set      210 315
Hilbert, David      219 220
Hilbert, irreducibility theorem of      182 198 199 207
Hillman, Abraham P.      120 179
Identity theorem for power series      183
Imaginary part      47
Index of a subgroup      133
Intermediate fields      80
Intermediate fields, finite number      80 150 261 294
Intersection of fields      23 223
Intestinal fortitude      118
Invariant subgroup      147
Irreducibility, of $x^{p} - A$      165 300
Irreducibility, of a polynomial in one variable      61
Irreducibility, of a polynomial in several variables      186
Irreducibility, of cyclotomic polynomials      119 120 277
Irreducibility, over $\mathbf{Q}$      64
Irreducibility, over $\mathbf{R}$      80 261
Irreducible polynomial, no multiple roots      71 178 254
Isomorphism of a field      290
Jacobson, Nathan      58 120 121
K-cycle      167
Kiernan, B. Melvin      7 179
Klein, Felix      58
Kline, Morris      7 178
kronecker      178
Kronecker's criterion      207 210
Kronecker's specialization      204 210 314
Lagrange, Joseph      5
Lagrange, theorem of      132
Landau, Edmund      57 120 121
Lang, Serge      220
Leading coefficient      38
Levinson, Norman      58 220
Lindemann, F.      3 58
Line in a field $\mathbf{F}$      16
M-set      108
Mac Lane, Saunders      58 179
Mann, W. Robert      220
McCoy, N.H.      120
Minimal polynomial      73
Modulo n      98
Modulus, of a complex number      55 86
Monic polynomial      38
Multiple extension      71 72
Multiple root      35
Multiple root, derivative at      39 240
Multiple root, impossible for irreducible polynomials      71 178 254
Nering, Evar D.      120
Niven, Ivan      58 120
Normal extension      141
Normal extension, equivalent properties      142
Normal subgroup      147
nth roots of unity      84
Olmsted, John M.      220
Order, modulo n      102 272
Order, of a group      131
Ore, Oysten      6
Period of an m-set      108
Permutation      40 167
Permutation, even and odd      47 242
Permutation, sequence of transpositions      46 242
Pi ($\pi$), transcendence of      48
Pierpont, James      178 220
Plane of a field $\mathbf{F}$      16
Polar representation      87
Pollard, Harry      58
Polynomial, in one variable      34
Polynomial, in several variables      39
Power series      182
Primes, infinite number      56 248
Primitive polynomial      71 254
Primitive roots modulo p      101
Primitive roots of unity      85
Primitive roots of unity, degree of      91 92
Proper divisor      61
Puiseux expansions      220
Quadratic equation      176 304
Quadratic extension      13 94
Quadrupling the cube      26 232
Quartic equation      126 177 281 307
Quartic equation, resolvent cubic      282
Quintic equation      172 173 301
Quintisection of an angle      31 84 98 233 262
Quotient      60
Radical extension      124
Radius of convergence      182
Rational functions      188
Rational operations      12
Rational roots theorem      39 237
Real part      47
Real point of an algebraic curve      197 313
Reciprocal power series      185
Redheffer, Raymond      58 220
Regular 17-gon, constructibility      113
Regular pentagon, constructibility      98 119 270 273
Regular polygons, constructibility      3 82 92 104 175
Regular polygons, necessary condition for constructibility      97 119 277
Regular polygons, sufficient condition for constructibility      106
Regular values      186
Regular values at infinity      199
Regular values, existence of      187
Relatively prime polynomials      68
Relaxation      80
Remainder      60
Resolvent cubic      282
Robbins, Herbert      57
Robinson, Abraham      120
Root functions      186
Root functions, analytic      190
Root of a polynomial      34
Roots of complex numbers      84
Ruffini, P.      5 177
Seidenberg, A.      220
Simple extension      71 72
Smallest field with a given property      23 223
Smith, David Eugene      7 178
Solution by radicals      4 123
Solutions to problems      8 221
Solvability by radicals      124 126 152 210 218
Solvable equations      174
Solvable groups      152
Splitting field      80 124 142
Splitting field, degree of      126 260 280
Squaring the circle      2 31 55
Steinberg, R.      58
Stewart, Ian      179
Stromberg, Karl      58
Subgroup      132
Symmetric functions, elementary      41
Symmetric functions, fundamental theorem on      41
Symmetric group      169
Symmetric group, not solvable      171
Symmetric group, polynomials with      171 173 174 176 181 220 302 303 304
Symmetric polynomial      40
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