as a sum of four squares 255 



where either a is odd or c is not divisible by 8. But H has already 

 been proved for these cases so that we have proved H in the ease 

 when a, h and c are positive and in particular for the algebraic fields 

 arising from a cubic with three real roots. 



§ (4). If however a and h are not both positive, in which case 

 equation (1) has imaginary roots, it is clear from equation (2) that 

 if f and q are taken to satisfy the inequahty iq > f' and t is a 

 sufficiently large positive number, then x+ t can be expressed as the 

 quotient of two positive definite quadratics in x. Hence as a; + « 

 is also the root of a cubic of the type (1) where a, h and c are 

 positive, it is clear that E holds for x + t^ provided t is a sufficiently 

 large positive number — integral or fractional. 



We have now from equation (1) 



(a; + I) (a;2 - (^ + a) X + A;) = (- P - a| + ^- - 6) a; + c+ 1^ 



where k and ^ are arbitrary rational quantities. We take k so large 

 that the discriminant 4A; - (| + af is positive and not of the form 

 M (this is always possible as k need not be an integer, take it say, 

 a fourth of an integer), and so that - P - a| + ^' - 6 is also 

 positive. 



Suppose ^ is not negative and put 



t _ ^^ + ^ . (6) 



then li is positive and also li > I since 



|3 + rt|2 + ^^ + C > 



because the real root of the cubic (1) is positive. Also E holds for 

 a; + I if it holds f or a; + |i . 



Take now 1= 0, and find ^^Az- from li, 4 ••• with of course 

 suitable values of l^, l\ ..., in the same way as |i was found 

 from f Then li, |,, 4 ... form a monotonic increasing sequence 

 whose limit is infinity. For if it were finite say L, then it follows 

 immediately from equation (6) that 



which is impossible as we have assumed that the cubic (1) has no 

 negative root. . . 



Hence after a certain stage {E) holds for x + $k. It is obvious 

 then from (5) that E holds for a-, so that we have also proved E for 

 the case of the field arising from a cubic with only one real root. 



