Mat 19, 1911] 



SCIENCE 



Ihl 



thus made to depend upon the analogous 

 problems for the galois corpora. 



After having shown how we may, by the 

 discussion of a congruence, form all the 

 ideals of given norm, Hilbert has sought a 

 new proof of the fundamental theorem of 

 Dedekind; he established it first for the 

 galois corpora and then easily extended it 

 to any corpus. 



Thus Hilbert was led to study the gen- 

 eral theory of galois corpora, and he intro- 

 duced a host of new notions, defining a 

 series of subcorpora, corresponding to dif- 

 ferent subgroups of the galois group of the 

 generating equation; these subgroups are 

 defined by certain relations they have with 

 any ideal prime of the corpus, and the 

 study of these subgroups opens for us 

 glimpses new and interesting of the struc- 

 ture of the corpus. 



Our author gave in 1896 a new proof of 

 Kronecker's theorem according to which 

 the roots of abelian equations can be ex- 

 pressed by the roots of unity. This demon- 

 stration purely arithmetical puts in evi- 

 dence the way of constructing all the 

 abelian corpora of a given group and dis- 

 criminant. 



But the works of Hilbert have had as 

 their principal object the study of corpora 

 relatively quadratic and relatively abelian. 



One of the essential points of the theory 

 of numbers is Gauss's law of reciprocity 

 in the subject of quadratic residues; we 

 know with what predilection the great 

 geometer returned to this question and how 

 he multiplied demonstrations. 



This law of reciprocity is capable of in- 

 teresting generalizations when we pass 

 from the domain of ordinary rational num- 

 bers to a domain of any rationality. Hil- 

 bert has succeeded in realizing this gener- 

 alization in the case where the corpus is 

 imaginary and has an odd number of 

 classes. He has introduced a symbol anal- 



ogous to that of Legendre, and the law of 

 reciprocity reached by him presents itself 

 in a simple form; the product of a certain 

 number of such sjrmbols must equal 1. 



This generalization presents all the more 

 interest since our author has succeeded in 

 showing that there are genera correspond- 

 ing to half of all the imaginable systems of 

 characters, a result which should be likened 

 to that of Gauss and which makes possible 

 the extension to a domain of any rational- 

 ity of this notion of the genus of quadratic 

 forms which is the subject of one of the 

 most attractive chapters of the "Disqui- 

 sitiones Arithmeticse. ' ' 



To go farther, Hilbert is obliged to in- 

 troduce a new notion and modify the defi- 

 nition of class. 



Two ideals belong to the same class in the 

 old or broad sense if their ratio is any ex- 

 isting algebraic number ; they belong to the 

 same class in the new or narrow sense if 

 their ratio is an existing algebraic number 

 which is positive as well as all its conju- 

 gates. The numbers of classes, whether 

 understood in the broad sense or in the nar- 

 row sense, are evidently in intimate rela- 

 tion and our author explains what the na- 

 ture of this relation is. But this new 

 definition allows Hilbert to express in 

 simpler language the theorems he had in 

 view. These theorems stated in their most 

 general form are, as Hilbert says, remark- 

 ably simple and of crystalline beauty; 

 their complete proof appeared to our au- 

 thor as the final aim of his studies on alge- 

 braic corpora. It is in this general form 

 we shall state them. 



If k is any corpus, there is a group Kk 

 which may be called its class corpus. Its 

 relative degree is equal to the number of 

 classes in the narrow sense. It is non- 

 ramified, that is to say, no ideal prime of k 

 is divisible by the square of an ideal prime 

 of Kk, and it contains all the non-ramified 



