758 



SCIENCE 



[N. S. Vol. XXXIII. No. 855 



corpora relatively abelian with regard to Tc. 



Its relative group is isomorphic to the 

 abelian group which defines the composi- 

 tion of the classes of ideals of k. 



The ideal primes of k, although prime in 

 relation to k, are not in general prime in 

 relation to Kk; they may, therefore, be 

 broken into factors ideal primes with re- 

 gard to Kk, the number of these factors 

 and the power to which they are raised, in 

 a word the mode of partition, depending 

 solely upon the class to which the ideal 

 considered belongs in the corpus k. 



Call "ambige" a number of Kk which is 

 positive as well as all its conjugates, and 

 which differs from these conjugates only 

 by a factor which is a complex unity. 



Each ambige of Kk corresponds to an 

 ideal of k and reciprocally. This property 

 is characteristic of the corpus Kk among 

 all the corpora relatively abelian with re- 

 gard to k. 



"We see the bearing of these theorems and 

 the light thrown on the notion of class, 

 since the mutual relations of classes of 

 ideals are reproduced as in a faithful pic- 

 ture by those of the algebraic integers of a 

 corpus. 



In reality Hilbert has completely proved 

 these theorems only in particular cases, 

 but these particular cases are very numer- 

 ous, exceedingly varied and broadly ex- 

 tended. He is, besides, he says, convinced 

 that his methods are applicable to the gen- 

 eral case. While sharing his conviction, 

 we must make reservation, so long as this 

 hope, legitimate as it may be, has not been 

 actually realized. 



We have spoken above of the law of reci- 

 procity relative to quadratic residues; we 

 must add that Hilbert has given an anal- 

 ogous law for his residues of any power, at 

 least for certain particular corpora. 



Summarizing, the introduction of ideals 

 by Kummer and Dedekind was an impor- 



tant advance; it generalized and at the 

 same time cleared up the classic results of 

 Gauss on quadratic forms and their compo- 

 sition. The works of Hilbert we have just 

 analyzed constitute a new step in advance, 

 not less important than the first. 



THEOKEM OP WARING 



Let us speak now of another entirely dif- 

 ferent arithmetical work. It pertains to 

 proving Waring 's theorem according to 

 which every integer can be broken into a 

 sum of N nth. powers, N depending only 

 upon n, just as, for example, it can always 

 be broken into a sum of four squares. 

 Needless to recall that this theorem up to 

 the present had simply been stated. 



What above all deserves to fix the atten- 

 tion in Hilbert 's proof is that it rests on a 

 new way of introducing continuous vari- 

 ables into the theory of numbers. 



We start from an identity where a 

 25uple integral is equated to the mth power 

 of the sum of five squares. Breaking up 

 the domain of integration into smaller do- 

 mains so as to have a series of approximate 

 values of the integral, as if it were a ques- 

 tion of evaluating it by mechanical quad- 

 ratures, and by the methods of passing to 

 the limit familiar to our author, we reach 

 another identity 



where the vk's are rational positive num- 

 bers and the Y's linear functions of the 

 x's with integral coefficients. The coeffi- 

 cients r and those of the Y's, as also the 

 number of these linear functions, depend 

 only upon m. 



Up to this point we have not gone out of 

 algebra, if not in showing that the coeffi- 

 cients r and those of Y are rational. To 

 get further our author establishes a series 

 of lemmas whose statement is too compli- 

 cated to be here reproduced and which 



