Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree

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Courier Corporation, 1956 - 289페이지
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This well-known work covers the solution of quintics in terms of the rotations of a regular icosahedron around the axes of its symmetry. Its two-part presentation begins with discussions of the theory of the icosahedron itself; regular solids and theory of groups; introductions of (x + iy); a statement and examination of the fundamental problem, with a view of its algebraic character; and general theorems and a survey of the subject. The second part explores the theory of equations of the fifth degree and their historical development; introduces geometrical material; and covers canonical equations of the fifth degree, the problem of A's and Jacobian equations of the sixth degree, and the general equation of the fifth degree. Second revised edition with additional corrections.
 

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CHAPTER I
3
Preliminary notions of the grouptheory
6
The cyclic rotation groups
8
The group of the dihedral rotations
10
The quadratic group
14
The group of the octahedral rotations
16
The group of the icosahedral rotations
17
On the planes of symmetry in our configurations
21
Infinite groups of linear substitutions of a variable
139
Solution of the tetrahedral octahedral and icosahedral equa tions by elliptic modular functions
144
Formulie for the direct solution of the simplest resolvent of the sixth degree for the icosahedron
147
Significance of the transcendental solutions
148
PART II
151
CHAPTER I
153
Elementary remarks on the Tschirnhausian transformation Brings form
156
Data concerning elliptic functions
159

General groups of pointsFundamental domains
23
The extended groups
24
Generation of the icosahedral group
28
Generation of the other groups of rotations
29
CHAPTER II
31
On those linear transformations of x + iy which correspond to rotations round the centre
34
Homogeneous linear substitutionsTheir composition
37
Return to the groups of substitutionsThe cyclic and dihedral groups
39
The groups of the tetrahedron and octahedron
40
SEC PAQK 6 The icosahedral group
43
Nonhomogeneous substitutions Consideration of the ex tended groups
46
Simple isomorphism in the case of homogeneous groups of substitutions
48
Invariant forms belonging to a groupThe set of forms for the cyclic and dihedral groups
50
Preparation for the tetrahedral and octahedral forms
54
The set of forms for the tetrahedron
55
The set of forms for the octahedron
58
The set of forms for the icosahedron
60
The fundamental rational functions
63
Remarks on the extended groups
66
CHAPTER III
67
Reduction of the formproblem
69
Plan of the following investigations
72
On the conformable representation by means of the function zZ
74
March of the 2 z2 function in generalDevelopment in series
77
Transition to the differential equations of the third order
79
Connection with linear differential equations of the second order
81
Actual establishment of the differential equation of the third order for zZ
83
Linear differential equations of the second order for z and z
85
Relations to Riemanns function
87
CHAPTER IV
90
On the group of an algebraical equation
91
General remarks on resolvents
94
The Galois resolvent in particular
97
Marshalling of our fundamental equations
100
Consideration of the formproblems
103
ESC PAGE 7 The solution of the equations of the dihedron tetrahedron and octahedron
104
The resolvents of the fifth degree for the icosahedral equation
106
The resolvent of the rs
108
Computation of the forms t and W
111
The resolvent of the its
113
The canonical resolvent of the s
114
Connection of the new resolvent with the resolvent of the rs
116
On the products of differences for the MS and the Ys
117
The simplest resolvent of the sixth degree
119
Concluding remarks
122
CHAPTER V
124
Algebraically integrable linear homogeneous differential equa tions of the second order
132
Finite groups of linear substitutions for a greater number of variables
135
Preliminary glance at the theory of equations of the fifth degree and formulation of a general algebraical problem
137
On Hermites work of 1858
162
The Jacobian equations of the sixth degree
164
Kroneckers method for the solution of equations of the fifth degree
168
On Kroneckers work of 1861
171
Object of our further developments
175
THE CANONICAL EQUATIONS OF THE FIFTH DEGREE
176
CHAPTER II
178
Classification of the curves and surfaces
180
The simplest special cases of equations of the fifth degree
181
Equations of the fifth degree which appertain to the icosa hedron
183
Geometrical conception of the Tschirnhausian transformation
186
Special applications of the Tschirnhausian transformation
188
Geometrical aspect of the formation of resolvents
190
On line coordinates in space
193
A resolvent of the twentieth degree of equations of the fifth degree
195
Theory of the surface of the second degree
196
s10 PAGE 1 NotationThe fundamental lemma
201
Determination of the appropriate parameter X
204
Determination of the parameter p
208
The canonical resolvent of the icosahedral equation
209
Solution of the canonical equations of the fifth degree
211
Gordans process
215
Substitutions of the X psInvariant forms
218
General remarks on the calculations which we have to perform
220
Fresh calculation of the magnitude mi
221
Geometrical interpretation of Gordans theory
223
Algebraical aspects after Gordan
225
The normal equation of the rs
228
firings transformation
230
The normal equation of Hermite
231
xiv
233
CHAPTER IV
234
The substitutions of the AsInvariant forms
236
Geometrical interpretationRegulation of the invariant ex pressions
239
The problem of the As and its reduction
243
On the simplest resolvents of the problem of the As
246
The general Jacobian equation of the sixth degree
247
Brioschis resolvent
250
Preliminary remarks on the rational transformation of our problem
251
Accomplishment of the rational transformation
254
Grouptheory significance of cogredience and contragredience
257
Introductory to the solution of our problem
259
Corresponding formula
262
CHAPTER V
265
Accomplishment of our first method
267
Relations to Kronecker and Brioschi
277
Special equations of the fifth degree which can be rationally
285
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