320 THIRD REPORT— 1833. 



It cos 7^ T = ^ = cos a, and cos p. r = 9 x = cos »i a, 



then these roots are reducible to the form 



x,dx,d^x, . . . 9"-i X, 

 or, 



cos a, cos m a, cos m? a, . . . cos ?w"~* « : 



and if we suppose m to be a primitive root to the modulus 

 2« + 1, then all the roots 



cos a, cos m a, cos in'- a, . . . cos m"~' « 



will be different from each other, and cos m" a = cos a ; con- 

 sequently it will follow, since the roots of the equation (2.) are 

 of the form 



X, 5 X, ^^ X, . . . fl"-' X, 



where 6'^x = x, they will admit, in conformity with the preceding 

 theorems, of algebraical expression. 



Abel has given the general form of the expression for these 

 roots, which in this case are all real ; and their determination 

 will involve the division of a circle into 2 n equal parts, the 

 division of an assigned or assignable arc into 2 n equal parts, 

 and the extraction of the square root of 2 « + 1 ; a conclusion 

 to which Gauss had also arrived, though he has not given the 

 steps of the process which he followed for obtaining it*. If we 

 suppose 2 n = 2", we shall get the case of regular polygons of 

 gw+i ^ \ sides, which admit of indefinite inscription in circles 

 by purely geometrical means. It will follow from the same i-e- 

 sult that the inscription of a heptagon will depend upon that 

 of a hexagon, the trisection of a given angle, and the extraction 

 of the square root of 7. 



Poinsotf has given a very remarkable extension to the theory 

 of the solution of the binomial equation a;" — 1 = 0, by showing 

 that its imaginary roots may be considered in a certain sense 

 as the analytical representation of the whole numbers which 

 satisfy the congruence or equation 



^» - 1 = M (;?), 



whose modulus (a prime number) is p : thus, the imaginary 

 cube roots of 1, or the imaginary roots of ;r^ — 1 = 0, are 



~~ ~ -, , and the whole numbers 4 and 2, 



2 



* Disquisitiones ArithmeticeB , p. 651. 



t Journal de I'Ecofe Poly technique, cahier 18. 



