156 REPORT 1860. 



of Gauss, and which gives the solution of any Abelian equation, is equally 

 applicable to any Abelian congruence; i. e. to any completely resoluble con- 

 gruence of order m, the m roots of which (considered with regard to the 

 prime modulus p) may be represented by the series of terms 



r,#(r>f a (r)....f-»(r> 

 the symbol <j> denoting a given rational [fractional or integral] function. 

 And as we can always express the roots of an Abelian equation by radicals 

 (i.e. by the roots of equations of two terms), so also the solution of an Abelian 

 congruence depends ultimately on the solution of binomial congruences. 

 When, for any prime modulus, an Abelian equation admits of being con- 

 sidered as an Abelian congruence, so precise is the correspondence of the 

 equation and the congruence, that (as Poinsot has observed in a memoir 

 in which he has occupied himself with the comparative analysis of the equa- 

 tion ai n =l, and the congruence x n =l, mod p*) we may consider the ana- 

 lytical expression of the roots of the equation as also containing an expression 

 of the roots of the congruence ; and by giving a congruential interpretation f 

 to the radical signs which occur in that expression, we may elicit from it the 

 actual values of the roots of the congruence. An example taken from 

 Poinsot's memoir will render this intelligible j. The six roots of the equation 

 a; 7 — I 



. =0 are comprised in the formula 



x— 1 



l + V 



x=- 



5 +HH^ + §^? + HH^-i'# 



6 



where the signs + and — are to be successively attributed to V — 7, and 

 where the product of the two cube roots is + V — 7, or — V — 7, according 

 to the sign attributed to V — 7, Considering the equation as a congruence 

 with regard to the modulus 43, and observing that 



V^7= ±6, mod 43, V2l= ± 8, mod 43, 

 we obtain in the first place 



x = ^-f-^x/ie +JV 7 - 8, mod 43, 



and x==— g-f-^ x/22+ g ^/ — % mod 43, 



the product of the two cube roots being congruous to 4-6 in the first formula, 

 and to —6 in the second ; and finally, observing that 



v/16 = 21, — 3, —18, mod 43, 



Z/^8 = 14,-2,-12, mod 43, 



1^22 = — 15, — 4, 19, mod 43, 



^/32 =+ 9, —20, +11, mod 43, 



sect. 3 (CEuvres, vol. i. p. 114, or Crelle, vol. iv. p. 131), and M. Serret's Algebre Superieure, 

 26th and 27th lessons. 



* "Sur l'Application de l'Algebre a la Theorie des Nombrcs," Memoires de l'Academie 

 des Sciences, vol. iv. p. 99. 



t Gauss employs the symbol \/ A, mod/;, to denote a root of the congruence *" = A, rnod^, 



r> 



just as he emplovs the symbol —, mod p, to denote the root of the congruence A.r=13, 



A 



mod p. The congruential radical f/\, mod /;, has of course as many values as the con- 

 gruence ;r n =A, mod p, has solutions; if that congruence be irrcsoluble, the symbol is im- 

 possible. 

 J See the memoir cited above, p. 125, 



