123 REPORT — 1860. 



u , «!, .. .m«-i denoting integral numbers, congruous or incongruous, mod q*." 

 A particular case of this theorem, relating to the equation =0 



(which may of course be regarded as the equation of the X— 1 periods, con- 

 sisting each of a single root), is due to Euler, and is included in his theory 

 of the Residues of Powers ; for it follows from that theory (see art. 12 of this 

 Report), that the binomial congruence a ,A — 1=0 (and therefore also the 



congruence x ~ 1 ^0, mod q) is completely resoluble for every prime of 



x — 1 

 the form ?w\+l. 



A remarkable relation subsists between the periods »j , r} x . . . n e -\ of the 

 equation F(y)=0, and the roots « , u v u 2 ... u e -\ of the congruence F(#)=0, 

 mod q. This relation is expressed in the following theorem : — 



" Every equation which subsists between any two functions of the periods, 

 will subsist as a congruence for the modulus q when we substitute for the 

 periods the roots of the congruence F(y)=0 taken in a certain order." 



It is immaterial which root of the congruence we take to correspond to any 

 given root of the equation. But when this correspondence has once been esta- 

 blished in a single case, we must attend to the sequence which exists among 

 the roots of the congruence corresponding to the sequence of the periods. 

 When u , u lf . . . u e -\ are all incongruous, their order of sequence is deter- 

 mined by the congruences 



« 1 =0(i/ o ), «„=$(«!), .... u ==(j>(ue-i), mod q, 



which correspond to the equations 



*1 = *(>»„). Vz = $(%), Vo = <K>/e-l). 



and which are always significant, although the coefficients of ty are frac- 

 tional, because it may be proved that their denominators are prime to the 

 modulus q. When u , u x , ...u e -i are not all incongruous [an exceptional 

 case which implies that q divides the discriminant of F(y)], a precisely simi- 

 lar relation subsists, though it cannot be fixed in the same manner, and though 

 the number of incongruous solutions of the congruence is not equal to the 

 number of the periods. (See a paper by M. Kummer in Crelle's Journal, 



* This theorem was first given by Schoeueinann (Crelle, vol. xix. p. 306) ; his demonstra- 

 tion, however, supposes that q ^.e,— a limitation to which the theorem itself is not subject. 

 The following proof is, with a slight modification, that given by M. Kummer (Crelle, vol. xxx. 

 p. 107, or Liouville, vol. xvi. p. 40d). From the indeterminate congruence of Lagrange (see 

 art. 10 of this Report) 



x(x-l) (x— 2) . . . . (x-q+\)^x1 -x, mod q, 

 it follows that 



(y-«?*)(y->jft-i)(y-'jft-2)..-(y-»»ife-?+i)=(y-ift) s -(y-fl*) 



=y ? - v/c 9 - (y - »?*) =y 7 - y> mod ?> 



observing that ';^=»JA-+Ind 9 > and that > if In<i ? be divisible hy e (or, which is the same 

 thing, if q satisfy the congruence qf=\, mod X), Jj/ f+ i a d ? = ''/,- -Multiplying together the 

 e congruences obtained by giving to k the e values of which it is susceptible in the formula 



(y-'u-)(y-'u-i)(y-'u--2)---(y-'u-!?+i)-y' ? -y> mod ?. 



we find 



F(y)F(y-l)F(y-2)...F(y- ? +l):=G/'-y)' ) raod?; 

 whence, by a principle to which we shall have occasion to refer subsequently (see Art. 69 ), it 

 appears that F(y) is congruous for the modulus q to a product of the form 



(y-««o)(y-«i) •••(y- M e-i)' 



