Mr Todd, A particular case of a theorem of Dirichlet 111 



A particular case of a theore^n of Dirichlet. By H. Todd, B. A., 

 Pembroke College. (Communicated, with a prefatory note, by 

 Mr H. T. J. Norton.) 



[Received 14 June 1917.] 



[The following note is an extract from an essay submitted to 

 the Smith's Prize Examiners. 



It will, perhaps, be convenient if I preface Mr Todd's argument 

 by explaining its relation to the theory of algebraical numbers. 

 The principal theorem is a famous one of Dirichlet's on the unities 

 of an algebraic corpus or order. It will be remembered that if ^ 

 is a root of an irreducible equation of the nth degree, the coefficients 

 of which are integers, then, if the coefficient of the nth. power of 

 the unknown is 1, ^ is an algebraic integer, and if in addition 

 the absolute term is + 1, ^ is a unity ; and further, that if ^ is an 

 integer of the ?ith degree, then the order of '^ is the aggregate of 

 numbers w of the form 



JUq ~r~ ^1 ^j *T • • • *^"ii 1 'J ) 



where x^... x^-i are rational whole numbers, every member of the 

 order of ^ being an integer of the wth or some loAver degree. 

 Dirichlet's theorem*, as modified by Dedekind and others, asserts 

 that if the irreducible equation satisfied by ^ has r real and 2s 

 imaginary roots, then the order of ^ contains r + s — 1 fundamental 

 unities, e^, .... e,.+^.._i , which are such that every unity contained in 

 the order is expressible in one and only one way as a product 



' ^ r+s-l' 



M'here t; is a root of unity contained in the order and m^, ... , m,.+.,_i 

 are rational integers ; and that, conversely, every such product 

 is a unity and a member of the order. The simplest cases of 

 this theorem are those in which the equation satisfied by "^ is 

 (i) a quadratic with two imaginary roots, (ii) a quadratic with two 

 real roots, (iii) a cubic with one real and two imaginary roots and 

 (iv) a quartic of which all the roots are imaginary. In the first 

 case, and in this alone, there are only a finite number of unities in 

 the order, and they are all roots of unity ; in the other cases 



* The theorem, when stated completely, has a wider scope, corresponding to a 

 wider definition of an ' order ' than is given above : what is there defined is more 

 properly called a 'regular order'. A general statement and proofs are given in 

 Bachmann, Zahlentheorie, vol. v., eh. 8. 



