250 Mr Mordell, On the representation of algebraic numbers 



On the representation of algebraic numbers as a sum of four 

 squares. By L. J. Mordell. (Communicated by Professor H. F. 

 Baker.) 



[Received 24 July 1920. Read 25 October.] 



Professor Landau in a recent paper* entitled "Uber die Zer- 

 le^ung total positiver Zahlen in Quadrate " states that about twenty 

 years ago, Professor Hilbertf gave without proof the theorem that 

 "Every number in an algebraic field (provided that neither it nor 

 any of its conjugate numbers are negative real quantities or if 

 all of the conjugate fields are imaginary) can be expressed as 

 the sum of the squares of four numbers of the field." This is an 

 extension of the well-known theoremj due to Fermat and proved 

 by Lagrange, that every positive integer§ can be expressed as the 

 sum of the squares of four other integers. From this result it 

 immediately follows that every positive fraction can be expressed 

 as the sum of the squares of four other fractions, a theorem 

 included in Hilbert's theorem, which is of course only a very special 

 case II in the arithmetical theory of quadratic forms with co- 

 efiicients in a given algebraic field. The development of this theory 

 however, is a matter of great difficulty, if only from the fact that it 

 requires a knowledge of the laws of quadratic reciprocity in the 

 field, the investigation of which, in even the simplest general cases, 

 requires a lengthy and detailed although very interesting dis- 

 cussion^. 



Professor Landau gives a simple proof for the quadratic field 

 based upon elementary algebra. He states however that he does not 

 know whether the theorem holds universally. This indicates that 

 a proof of Hilbert's general theorem is not easy, a view which 

 is confirmed by considering the general theory underlying the 

 question. The following proof for a cubic field may therefore be of 

 interest**. 



§ (1), Let then x be the root of an irreducible cubic equation 



x^ — ax^ -f bx — c = ( 1 ) 



* Nachrichten der K. Gesellschaft der Wissenschaften zu Gottingen, 1919, pp. 392- 

 396. 



t Grundlagen der Geometric, § (38). 



j Bachmann, ZaMentheorie, vol. 4, p, 151. 



§ The terms integers, fractions, etc. refer to rational quantities unless otherwise 

 stated. 



II See also an account of some of A. Meyer's work on equations of the form 

 ax^ + by^ + cz^ + dt^ + eu^ = in Bachmann, ZaMentheorie, vol. 4, pp. 259, 266. 



^ See for example Hilbert, Ueber die Theorie des relativquadratischen Zahlkorpers, 

 Mathematische Annalen, vol. 51, pp. 1-127. An elementary introduction is given 

 in Sommer " Vorlesungen iiber Zahlen theorie," section 5, of which there is a French 

 translation. 



** I am very greatly indebted to Professor Landau for having gone through this 

 paper and for having suggested a number of improvements ia the exposition. 



