## Lectures on the Icosahedron and the Solution of Equations of the Fifth DegreeThis 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. |

### ´Ù¸¥ »ç¶÷µéÀÇ ÀÇ°ß - ¼Æò ¾²±â

¼ÆòÀ» Ã£À» ¼ö ¾ø½À´Ï´Ù.

### ¸ñÂ÷

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 |

### ±âÅ¸ ÃâÆÇº» - ¸ðµÎ º¸±â

Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree Felix Klein ÂªÀº ¹ßÃé¹® º¸±â - 1956 |

### ÀÚÁÖ ³ª¿À´Â ´Ü¾î ¹× ±¸¹®

accessory algebraical Annalen arbitrary auxiliary equation binary binomial equation Brioschi canonical equation canonical resolvent canonical surface co-ordinates coefficients collineations compute connection consider consideration construct corresponding covariant curve cyclic groups denote determined developments diagonals differential equation dihedral group dihedron elliptic functions elliptic modular functions equa equal exposition expression fact factor fifth degree form-problem formula func fundamental fundamental domains further Galois Galois group geometrical given groups of linear hedral Hence Hermite Herr Kronecker homogeneous functions icosa icosahedral equation icosahedral substitutions icosahedron identical integral functions introduce invariant forms irrationalities isomorphic Jacobian equation linear substitutions magnitudes Math means method modular equation obtain octahedron operations paragraph parameter permutations plane preceding chapter problem quadratic group rational function relation remain unaltered remark respectively rotations simply sixth square root sub-group summits tetrahedral group tetrahedron theorem theory of equations third order tions Tschirnhausian transformation Ueber Unabridged republication values variables zv z2