ON THE THEORY OP NUMBERS. 255 



If p and q are both real primes, it is easily seen that either of them is a 

 quadratic residue of the other in the complex theory, or = = ^ 



But, as j9 may or may not be a quadratic residue of g in the theory of real 

 integers, we see that the values of the symbols <- and l™ \ are not neces- 

 sarily identical. 



This theorem is only enunciated in Gauss's memoir (Art. 60), and, as he 

 speaks of it as a special case of the corresponding theorem for biquadratic 

 residues, it is probable that his demonstration of it was of the same nature 

 with that which he had found of the law of biquadratic reciprocity. How- 

 ever, a simple proof of it, depending on Legendre's law of reciprocity, has 

 been given by Dirichlet in Crelle's Journal*. He shows that, if q be a prime 



of the first kind, pL±£f| =/ c L+i^\; and that, if a + bi be any prime of the 



second kind in which b is even, | , : — I « » )■ The law of recipro- 



\_a + bij \ar + b l J 



city is easily deducible from these transformations. If, for example, a + bi, 

 a + fti, be primes of the second species in which both b and ft are even, we 

 have simultaneously 



ra+ftf] _( aa-tbft \ r a + bi l _/ aa + bft \ 

 [.a + bi] \ p P U+flJ V * / 



where p = N (a + bi) ; «r = N (a + jSt). But fe2±££\ = (^jr\ b Y 



Jacobi's formula (see Art. 17 supra); and ( ) = ( — TT75r ^ s0 



pvs=(aa+bft) 2 + (aft— ba) 2 ; whence we infer/ P™ \=\, or ' wluch is the 



thing, ( * W-gQ ; and therefore finally, [ a -±B = fS±Sl 

 b \aa+bft) \aa + bftj 3 \a + bi] La + ftiJ" 



The complementary theorems which have to be united with this formula 



rad~ ( 1} ' \J+iftr { 1} 8 



(see Dirichlet, Crelle, vol. xxx. p. 312); and they, as well as the formula of 

 reciprocity itself, admit of an extension similar to that which Jacobi has 

 given to the corresponding formulae of Legendre. 



28. Reciprocity of Biquadratic Residices. — We now come to the theorem 

 which first suggested the introduction of complex numbers. 



If p be any (complex) prime, and k be any residue not divisible by p, we 



denote by ( — ) the power i e of i, which satisfies the congruence ki^P-V^i . 



It will be observed that when p is a prime of the second species, the quadri- 

 partite classification of the real residues of p which we thus obtain is identical 

 with that which we obtain for N/> in the real theory (see Art. 24 supra) ; for 

 the numbers/ and — / being the roots of the congruence ar + 1=0, mod Np, 



* Crelle, vol. is. p. 379. 



same 



are 



