Theory of substitutions and its applications to algebra

Title page......Page 1
PREFACE......Page 3
TRANSLATOR'S NOTE......Page 5
TABLE OF CONTENTS......Page 7
ERRATA......Page 12
1-3. Symmetric and single-valued functions......Page 13
4. Elementary symmetric functions......Page 16
5-10. Treatment of the symmetric functions......Page 17
12. Euler's formula......Page 22
13. Two-valued functions; substitutions......Page 24
14. Decomposition of substitutions into transpositions......Page 25
15. Alternating functions......Page 26
16-20. Treatment and group of the two-valued functions......Page 27
22. Notation for substitutions......Page 30
24. Their number......Page 33
25. Their applications to functions......Page 34
26-27. Products of substitutions......Page 35
28. Groups of substitutions......Page 37
29-32. Correlation of function and group......Page 39
34. Symmetric group......Page 44
35. Alternating group......Page 45
36-38. Construction of simple groups......Page 48
39-40. Group of order $p^f$......Page 52
41-44. Relation of the order of a group to the number of values of the corresponding function......Page 56
45. Groups belonging to the different values of a function......Page 60
46-47. Transformation......Page 61
48-50. The Cauchy-Sylow Theorem......Page 63
51. Distribution of the elements in the cycles of a group......Page 66
52. Substitutions which belong to all values of a function......Page 69
53. Equation for a $\\rho$-valued function......Page 72
55. Discriminants of the functions of a group......Page 74
56-59. Multiple-valued functions, powers of which are single-valued......Page 76
60-61. Simple transitivity......Page 82
62-63. Multiple transitivity......Page 84
64. Primitivity and non-primitivity......Page 86
65-67. Non-primitive groups......Page 87
69-71. Commutative substitutions; self-conjugate subgroups......Page 90
72-73. Isomorphism......Page 94
74-76. Substitutions which affect all the elements......Page 96
77-80. Limits of transitivity......Page 99
81-85. Transitivity of primitive groups......Page 103
86. Quotient groups......Page 107
87. Series of composition......Page 108
88-89. Constant character of the factors of composition......Page 109
91. Construction of compound groups......Page 113
92. The alternating group is simple......Page 114
93. Groups of order $p^\\alpha$......Page 115
94. Principal series of composition......Page 116
95. The factors of composition equal prime numbers......Page 119
97-98. The degree and order equal......Page 121
99-101. Construction of isomorphic groups......Page 123
103-105. Functions belonging to the same group can be rationally expressed one in terms of another......Page 126
106. Families; conjugate families......Page 130
107. Subordinate families......Page 131
108-109. Expression of the principal functions in terms of the subordinate......Page 132
110. The resulting equation binomial......Page 135
111. Functions of the family with non-vanishing discriminant......Page 137
112. Special cases......Page 140
113. Change in the form of the question......Page 141
114-115. Functions whose number of values is less than their degree......Page 143
116. Intransitive and non-primitive groups......Page 144
117-121. Groups with substitutions of four elements......Page 145
122-127. General theorem of C. Jordan......Page 150
128. Preliminary theorem......Page 156
129. Groups $\\Omega$ with $r=n=p$. Cyclical groups......Page 157
130. Groups $\\Omega$ with $r=n=pq$......Page 158
131. Groups $\\Omega$ with $r=n=p^2$......Page 160
132-135. Groups which leave, at the most, one element unchanged. Metacyclic and semi-metacyclic groups......Page 161
136. Linear fractional substitutions. Group of the modular equations......Page 165
137-139. Groups of commutative substitutions......Page 168
141. Condition for the denning function......Page 172
143. Arithmetic substitutions......Page 174
144. Geometric substitutions......Page 175
146-147. Order of the linear group......Page 177
148. The equations of the second degree......Page 180
149. The equations of the third degree......Page 181
150. The equations of the fourth degree......Page 182
152. The general problem formulated. Galois resolvents......Page 183
153-154. Affect equations. Group of an equation......Page 184
156. Fundamental theorems on the group of an equation......Page 188
157. Group of the Galois resolvent equation......Page 189
158-159. General resolvents......Page 190
161. Definition and irreducibility......Page 192
162. Solution of cyclic equations......Page 193
163. Investigation of the operations involved......Page 195
164-165. Special resolvents......Page 196
166. Construction of regular polygons by ruler and compass......Page 199
167. The regular pentagon......Page 200
168. The regular heptadecagon......Page 201
169-170. Decomposition of the cyclic polynomial......Page 205
171-172. One root of an equation a rational function of another......Page 209
173. Construction of a resolvent......Page 211
174-175. Solution of the simplest Abelian equations......Page 212
176. Employment of special resolvents for the solution......Page 214
177. Second method of solution......Page 215
178-180. Examples......Page 216
181. Abelian equations. Their solvability......Page 222
182. Their group......Page 223
183. Solution of the Abelian equations; first method......Page 224
184-186. Second method......Page 227
188-189. Examples......Page 230
190-193. Groups analogous to the Abelian groups......Page 234
194. Equations all the roots of which are rational functions of two among them......Page 237
196. Their group in the case $n=p$......Page 238
197. The binomial equations......Page 239
199. Triad equations......Page 241
200-201. Constructions of compound triad equations......Page 242
202. Group of the triad equation for $n=7$......Page 244
203-205. Group of the triad equation for $n=9$......Page 246
206. Hessian equation of the ninth degree......Page 250
207-209. Rational domain. Algebraic functions......Page 252
210-211. Preliminary theorem......Page 255
212-216. Roots of solvable equations......Page 257
217. Impossibility of the solution of general equations of higher degrees......Page 262
218. Representation of the roots of a solvable equation......Page 263
219. The equation which is satisfied by any algebraic expression......Page 266
220-221. Changes of the roots of unity which occur in the expressions for the roots......Page 268
222-224. Solvable equations of prime degree......Page 272
226. Definition of the group......Page 278
228. Its primitivity......Page 279
229. Galois resolvents of general and special equations......Page 281
230. Composition of the group......Page 284
231. Resolvents......Page 286
232-234. Reduction of the solution of a compound equation......Page 287
236-238. Adjunction of the roots of a second equation......Page 291
239-241. Criteria for solvability......Page 298
242. Applications......Page 300
243. Abel's theorem on the decomposition of solvable equations......Page 302
244. Equations of degree $p^k$; their group......Page 305
245. Solvable equations of degree $p$......Page 307
246. Solvable equations of degree $p^2$......Page 309
248-249. Expression of all the roots in terms of a certain number of them......Page 311