G Art, 5. -T. Tiikenoiiclù : 



§. 3. 



Let us now introduce a new function: 



./ -, H«) . 



■ '-"^ = W ' 



the ambiguity arising from the cube root of g^ Avil be remo^'ed 

 afterwards. If we suppose that <« is relatively prime to 3, then 



where ii^(:i') and F,,(.c) denote rational integral functions of the 

 forms 



i?/,(aO = x"' + a^x""^+ +a;X""^'-'+ + cim-ix, 



P,(rr) = f_{^x—' + h,x"'-'+ +b,x'-"'-'+ + {-iy"-\o-^'+'\ 



with integral coefficients in Z(/0- 



Hence, if we regard t(){) as an unknown quantity-, the equa- 

 tion 



Iir--(fm)P, = 



is of the ??zth degree, and its absolute term is associated with -(/m). 

 All the roots of this equation are given by 



where ^ is to take all the values of the coinplete system of residues 

 with respect to tlie modulus /^. Hence we obtain the relation 



the sym1)ol =^ meaning "is associated with." 



Xow, this last relation exactly corresponds to that of tlie same 

 form, which presents itself in the comiilex nuiltiplication of the 

 function sn." Hence, starting from this relation, we can deduce the 

 the folloning results: '' 



1) Weber: Algebra III, §. 157 i22i. 



2) Weber: Algebra III, §. 157. 



