756 



SCIENCE 



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



We see under what a new and elegant 

 aspect present themselves to-day, thanks to 

 Hilbert, problems so many geometers had 

 for fifty years attempted. 



THE NUMBER 6 



Hermite was the first to prove that the 

 number e is transcendant, and shortly after- 

 ward Lindemann extended this result to 

 the number tt. 



This was a victory important for sci- 

 ence, but Hermite 's methods were still sus- 

 ceptible of betterment; however ingenious 

 and however original they were, one felt 

 they did not lead to the goal by the short- 

 est way. This shortest way Hilbert has 

 found, and it seems that henceforth no one 

 can hope to give new simplification to the 

 proof. 



This was the second time that Hilbert 

 had given, of a theorem known but only es- 

 tablished by means most arduous, a proof 

 of astonishing simplicity. This faculty of 

 simplifying what had seemed at first com- 

 plex thus presents itself as one of the char- 

 acteristics of his genius. 



ARITHMETIC 



The arithmetical works of Hilbert per- 

 tain principally to algebraic corpora. The 

 aggregate of numbers which can be ex- 

 pressed rationally as functions of one or 

 several algebraic numbers constitutes a do- 

 main of rationality, and the aggregate of 

 the numbers of this domain which are 

 algebraic integers constitutes a corpus. If 

 we consider then all the algebraic numbers 

 of a corpus which can be put under the 

 form 



where the a's are given numbers of the 

 corpus, and the x's indeterminate numbers 

 of the same corpus, the aggregate of these 

 numbers is what is called an ideal. That 



which gives interest to this consideration is 

 that the ideals obey in what concerns their 

 divisibility the ordinary laws of arithmetic 

 and that in particular every ideal is decom- 

 posable in one way and only one into ideal 

 primes. This is the fundamental theorem 

 of Dedekind. 



On the other hand, we may consider 

 numbers which satisfy an algebraic equa- 

 tion of which the coefficients belong to a 

 domain D of rationality. These numbers 

 and those rationally expressible by means 

 of them define a new domain of rationality 

 D' more extended than D; and an alge- 

 braic corpus E' more extended than the 

 corpus K which corresponds to D. We 

 then may relate the corpus K', not to the 

 ordinary rational numbers and to the cor- 

 pus of the integers of ordinary arithmetic, 

 but to the domain D and to the algebraic 

 corpus E. We then may speak of the rela- 

 tive degree of E' with reference to E, of 

 the relative norm of an algebraic number 

 of E' with reference to K, etc. There will 

 be corpora relatively quadratic obtained 

 by the adjunction to the domain D' of a 

 radical V;^, M being a number of the do- 

 main D, and corpora relatively abelian, 

 obtained by the adjunction to D of the 

 roots of an abelian equation. This is a sort 

 of generalization of the ideas of Dedekind, 

 that Hilbert is doubtless not the first to 

 have seen, but from which he has drawn 

 an unexpected advantage. 



We should also speak of galois corpora, 

 whose generating equation is a galois equa- 

 tion. Any corpus is contained in a galois 

 corpus, in the same way the corpus E of 

 which we have just spoken is contained in 

 the corpus E' ; and this galois corpus is 

 easily obtained by adjoining to the domain 

 of rationality, not only one of the roots of 

 the generating algebraic equation of E, but 

 all its roots. 



Questions relative to any corpus are 



