as a sum of four squares 261 



where there is no loss of generaHty in supposing that a, b, c and al] 

 the rational numbers dealt with in this paper are integers, except 

 when obviously otherwise. Any number/ in the cubic field can be 

 expressed in the form 



Df = Ax^ + Bx + C 

 where A,B,C and D are integers, of which D is positive. Hilbert's. 

 theorem then asserts that 



where 



A/i = A^^ + ^1^ + Cj^, etc. 

 and A^, B^, etc. are integers, can be found so that 



provided that neither/ nor any of its two conjugate numbers are 

 negative real quantities. We shall refer shortly to this theorem by 

 saying that H holds for/. 



The case when / = can be dismissed at [once by noting that 

 = 02 + 02 + 02 + 02. ' 



§ (2). If H holds for two numbers F^,F^, it also holds for their 

 product since as is well known 



(/l' +72^ +/3' +f^) i<l>l' + <f>2' + <t>Z^ + <f>^) = 01^ + 'As' + 'As' +'A4' 



where ^Ai = Mi + A-^s + Mz + h<t>^ etc. ; 



and also for their quotient since 



FJF,^F,FJF,\ 



In particular if n is any positive rational fraction, H holds for 

 w/if it holds for/. 



Obviously H holds for the number x^ + g if ^ is a positive 

 integer which can be expressed as the sum of the squares of three 

 integers. This is always* the case except when q is of the form 

 4^(8w+7); and we shall say for shortness that any number, 

 positive or negative, of the form 4^ (8m + 7) is of the form M. 

 From the definition it is necessary but not sufficient that 



31 = 0, 4, or 7 (mod 8) (A). 



In particular the highest power of 2 dividing M must have an 

 even exponent. Similar results hold for x^ + q when §• is a fraction 

 (positive of course) 



X/fl — A/x//x2 



such that Xfi is not of the form M, and we can still say in this case 

 that the fraction q is not of the form M. It is also clear that H holds 

 for 



ax^+ bx + c = [{2ax + bf + iac - b^^ia 



* Bachmann, Zahlentheorie, vol. 4, p. 146| 



VOL. XX. PART II. 17 



