ON THE THEORY OF NUMBERS. 121 



where a is an imaginary 5th root of unity. Precisely similar remarks apply 

 to the theories of residues of 8tli and 12th powers, — real primes of the forms 

 8«+l, 12« + 1> resolving themselves into four factors composed of Sth and 

 12th roots of unity respectively. By considerations similar to those pre- 

 viously employed by him in the case of biquadratic and cubic residues, 

 Jacobi succeeded in demonstrating (though he has not enunciated) the for- 

 mulae of reciprocity affecting those powers for the particular case in which 

 one of the two primes compared is a real number. But it would seem that 

 he never obtained the law of reciprocity for the general case of any two 

 complex primes; and indeed, for a reason which will afterwards appear, it 

 was hardly possible that he should do so, so long as he confined himself to 

 the consideration of those complex numbers which present themselves in the 

 theory of the division of the circle. No less unsuccessful were the efforts of 

 Eisenstein to obtain the formulae relating to 8th powers, by an extension of 

 the elliptical properties employed by him in his later proofs of the biqua- 

 dratic theorem*. It does not appear that any subsequent writer has occu- 

 pied himself with these special theories ; while, on the other hand, the theory 

 of complex numbers composed with roots of unity of which the exponent is 

 any prime, has been the subject of an important series of investigations by 

 MM. Diriehlet and Kummer, and has led the latter eminent mathematician 

 to the discovery and demonstration of the law of reciprocity, which holds for 

 all powers of which the exponent is a prime number not included in a cer- 

 tain exceptional class. 



40. Necessity fur the Introduction of Ideal Primes. — The fundamental pro- 

 position of ordinary arithmetic, that if two numbers have each of them no 

 common divisor with a third number, their product has no common divisor 

 with that third number, is, as we have seen, applicable to complex num- 

 bers formed with 3rd or 4th roots of unity, because it is demonstrable that 

 Euclid's theory of the greatest common divisor is applicable in each of those 

 cases. With complex numbers of higher orders this is no longer the case; 

 and it is accordingly found that the arithmetical consequences of Euclid's 

 process, which are of so much importance in the simpler cases, cease to exist 

 in the general theory. In particular, the elementary theorem, that a number 

 can be decomposed into prime factors in one way only, ceases to exist for 

 complex numbers composed of 23rd f or higher roots of unity — if, at least 

 (iu the case of complex as of real numbers), we understand by a prime fac- 

 tor, a factor which cannot itself be decomposed into simpler factors %. It 

 appears, therefore, that in the higher complex theories, a number is not 

 necessarily a prime number simply because it cannot be resolved into com- 

 plex factors. But by the introduction of a new arithmetical conception — 

 that of ideal prime factors — M. Kummer has shown that the analogy with 

 the arithmetic of common numbers is completely restored. Some prelimi- 

 nary observations are, however, necessary to explain clearly in what this con- 

 ception consists. 



* See M. Kummer, " Ueber die Allgemeinen Reciprocitiitsgesetze," p. 27, in the Memoirs 

 of the Berliu Academy for 1859. 



t For complex numbers composed with 5tli or 7th roots of unity, the theorem still exists ; 

 for 23 and higher primes it certainly fails ; whetherlt exists or not for 11, 13, 17, and 19, 

 has not been definitely stated by M. Kummer (see below, Art. 50). 



.t "Maxime dolendum videtur" (so said M. Kummer in 1814) "quod hsec numerorum 

 reahum virtus, ut in factores primos dissolvi possint, qui pro eodem numero semper iidem 

 Bint, non eadem est numerorum complexorum, qua si esset, tota hsec doctrina, quse magnis 

 adhuc difficultatibus premitur, facile absolvi et ad linem perduci posset." (See his Disserta- 

 tion in Liouville's Journal, vol. xii. p. 202.) In the following year he was alreadv able to 

 withdraw this expression of regret. 



