154 REPORT — 1860. 



8, while the square of every ideal number (occurring as a factor of a real 

 prime inferior to 1000) is actual. The methods employed by Dr. Reuschle 

 in the calculation of his tables have not yet been published by him. In 

 some instances, as M. Kummer has observed, they have not led him to the 

 simplest possible forms of the ideal primes. 



A particular investigation relating to the ideal factors of 47, composed of 

 23rd roots of unity, has been given by Mr. Cayley (Crelle, vol. lv. p. 192, 

 and lvi. p. 186). 



64. The investigations relating to Laws of Reciprocity, which have so long 

 occupied us in this report, have introduced us to considerations apparently 

 so remote from the theory of the residues of powers of integral numbers, that 

 it requires a certain effort to bear in mind their connexion with that theory. 

 It will be remembered that the complex numbers to which our attention has 

 been directed are not of that general kind to which we have referred in art. 41, 

 but are exclusively those which are composed of roots of unity. The theory 

 of complex numbers, in the widest sense of that term, does indeed present to 

 us an important generalization of the theory of the residues of powers; for 

 the theorem of Fermat (see art. 53) subsists alike for every species of com- 

 plex numbers. But the complex numbers of Gauss, of Jacobi, and of M. 

 Kummer force themselves upon our consideration, not because their proper- 

 ties are generalizations of the properties of ordinary integers, but because 

 certain of the properties of integral numbers can only be explained by a 

 reference to them. The law of quadratic reciprocity does not, as we have 

 seen, necessarily require for its demonstration any considerations other than 

 those relating to ordinary integers ; the real prime numbers of arithmetic are 

 here the ultimate elements that enter into the problem. But when we come 

 to binomial congruences of higher orders, we find that the true elements of 

 the question are no longer real primes, but certain complex factors, composed 

 of roots of unity, which are, or may be conceived to be, contained in real 

 primes. For we find that the law which expresses the mutual relation (with 

 respect to the particular kind of congruences considered) of two of these 

 complex factors is a primary and simple one ; while the corresponding rela- 

 tions between the real primes themselves are composite and derivative, and, in 

 consequence, complicated. It thus becomes indispensable, for the investiga- 

 tion of the properties of real numbers, to construct an arithmetic of complex 

 integers; and this is what has been accomplished by the researches, of which 

 an account has been given in the preceding articles. 



The higher laws of reciprocity (like that of quadratic residues) may be 

 considered as furnishing a criterion for the resolubility or irresolubility of 

 binomial congruences ; and this, though not the only application of which they 

 are susceptible, is that which most naturally suggests itself. When the bi- 

 nomial congruence is cubic or biquadratic, it is easy to resolve the real prime 

 modulus into factors of the {orma + bp,or a + bi (arts. 37 and 24-), and equally 

 easy to determine the value of the critical symbol of reciprocity by a uni- 

 form and elementary process (see art. 36). For these, therefore, as well as 

 for quadratic congruences, the criterion deducible from the laws of recipro- 

 city is all that can be desired. But for binomial congruences of higher 

 orders this criterion is not a satisfactory one, because of the difficulty of 

 obtaining the resolution of a real prime into its complex factors, and also 

 because of the impossibility of determining the value of the critical symbol 

 by the conversion of an ordinary fraction into a continued fraction. 



The only known criterion applicable to such congruences is the following, 

 the demonstration of which is deducible from the elements of the theory of 

 the residues of powers : — Let ai n =A, mod p, represent the proposed con- 



