ON THE THEORY OP NUMBERS. 265 



it as in the case of ordinary integers. The prime numbers to be considered 

 in this theory are — 



(1) Real primes, as 2, 5, 11, 17, &c. of the form 3ra+2. 



(2) Imaginary primes of the form A + Bp, having for their norms real 

 primes of the form 3w+ 1. 



(3) The primes 1— p, 1— p 2 , having 3 for their norm. 

 The units are +1, +p, and + p 2 . 



If A + Bp be any complex number not divisible by 1— p, it may be seen 

 that of the 3 pairs of numbers, + (A-f Bp), + p(A + Bp), +p 2 (A + Bp), 

 there is always one, and one only, which, when reduced to the form a + bp, 

 satisfies the congruences a = + 1, b=0, mod 3. Such a number is called a 

 primary number. The product of two primary numbers, taken negatively, 

 is itself primary. 



If a. be any prime of this theory, and k any number not divisible by a, 

 Fermat's Theorem is here represented by the congruence A Na-1 = l, mod a. 



Denoting by (-) that power p s of p which satisfies the congruence 

 * 4(Na-1) ==p», the law of cubic reciprocity is contained in the formula 



V/3/3 \a/3 



a and /3 denoting any two primary complex primes. 



The demonstration of this theorem follows quite naturally from the for- 

 mulae cited in Art. 30. Applying them to this particular case, we have, if/> 

 denote a real prime of the form 3w + l, 



(i) F(p).F(p 2 )=p, 

 00 [F(p)] 3 =^(p), 



(in) Kp)'W)=p> 



(iv) i//(yiO- '))=(), modp; 



from which we may infer that yiCP-^^p, mod »^(p). (Compare Art. 29.) 

 In the equation (iii), ^(p) and i|/(p 2 ) are primary ; for from the equation 

 [ f (p)V=pHp)> i* appears that i/,(p)== — 1, mod 3. The congruence 



y4<P-i)==p, mod i//(p), implies that (-l—\=p% whence if y°=k, mod p, 



F(p)= 2 (AW, 



*=i w» 



F(p 2 )= 2 (AW; 



k=l \pj» 



where p 1 =\p(p)- By these formulae the several cases of the theorem of reci- 

 procity may be proved, as follows*: — 



First, let q be a prime of the form 3rc + 2. Then 



k=p—\/ fo \q 



[F(p)]« == 2 ( — )««*, mod j, 



* Eisenstein in Crelle's Journal, vol. xxvii. p. 289. But in this, as in many of his earlier 

 researches, Eisenstein had been anticipated more than ten years by Jacobi. 



