53 Art. 5.— T. Takenonchi: 



Hence Ä"[6] is nothing else l)ut the elementary corpns \I, whose 

 relative discriminant is divisihle by 3. 



Let us now determine the nature of the relative discriminant 

 of Ä"[??2] , as a sequel to our preceding investigations. 



If Ave assume, as in §. 18, that 



VI =^y'i2)^'-i jj^'i 



then we liave 



K[iJi:]^K[p^qK[p!;q K[p^!q , 



where Z[p''i] = P-'Q^'O, if j;,=^'2 or ^^, = '2, //,r=],'2, 



and PX2'*0 > AT2'''] > P%2'''''), if //, > 2. 



But, the relative discriminant of F'Xp^-0 is a power of />„ except 

 when p^^' = 2 or 3. Hence the relative discriminant of 7v"[?«] must 

 contain all the prime factors of m, except in the following cases : 



(i) m=0 (mod. 2), but #;0 (mod. 2-), 

 (ii) w = (mod. 3), but ^0 (mod. 3-). 



On the other hand, since 



"\ve see tliat the relative discriminant of J'^lm] docs not contain 

 prime factors relatively prime to m. Thus, if we exclude the 

 above tAVO cases (i) and (ii), we may conclude that the relative 

 discriminant of Ä"[?>?] contains all the prime factors of ??z, but no 

 other. 



In case (i), since w is divisible by 2, the corpus K[7n] con- 

 tains the elementary corpus HI or VI (§. 21), except when ;«=2. 

 Therefore its relative discriminant must contain the factor 2. Thus 

 we see that this case (i) need not be excluded from tlie aboA'c state- 

 ment, provided that w=f-2. 



Next, consider the case (ii), and suppose that ??i=^3. If ^//i be 

 even, then K[m] contains VI, and consequently its relative dis- 

 criminant is divisible ])y 3. 



If, on the contrary, m be odd, then 7v [?«] does not contain 



VI itself, but a derived corpus of III,- (? = 1,2, ,_pi=/:3)and VI. 



Hence we have to inquire, whether the relative discriminant of 



