On the E-elatively Abelian Corpora. 65 



least all tJie integers of the form 4];)^ where j) is a natural j)t'hne (JJk 

 case i')=:2 he'imj included). 



Therefore all the relativehj Abelian covpora tvith resjiect to h{j>) 

 are exhausted by the division-corpora PyVi), and consefjnenth/ also In/ 

 Weber'' s division-corj^ora. 



When ni is odd, tlio relative discriminant of P'\m) does not 

 contain the factor 2 (§. -")). Therefore PX/n) cannot contain tlie 

 elementary corpora III. V, VI, VII. Hence we see that P\rn). 

 m being odd, is composed of 



(i) I,, IIj, IV, / = 1, 2, , excepting the vahie, 



for which -^ = 1+2//, 

 VIII,, ^Yhen m = (mod.-;), 



(ii) IX, when m=0 (mod. 3'-'), 



(iii) a derived corpus of III,- (/ = 1,2, ; III,=YI, if - = 1 +2o), 



(iv) the derived corpus of Y,- ( / = 1, 2, ; Y, = VII, if -,= 1 + 1f>). 



Bnt, as we l]a^'e remarked before, the corpns P\m). when 

 m is even, coincides with P{iii). whose constitution is given in (34) 

 and (35). 



Therefore we conclude that 



the relativehj Abelian corpora with respect to k(p) are exhausted 

 also by the diivision-corpora P\i)i). 



§. 21. 



Let us now proceed a step further and consider the constitu- 

 tion of the s^ra/^ /-corpus K\i){\. 

 As we have shewn in ^. 1<S, 



when ^) is an odd prime, or wrien^9 = 2, h = \, 2. In otlier cases, in 

 which j9=2, /i>2, the corpus PX'2!') is relatively quadratic with 

 respect to 7\'[2'']. 



Hence, if we remember the relation (30) and the value of 

 the relative degree 7/, the decomposition of K\_ni^ into elementary 

 corpora can be effected by §. 19 without any difficulty. 



