[fields] the existence THEOREM 9i 



and, what suffices for our purpose, we can evidently effect a modular 

 factorisation of f{z, u) in the form 



12. f(z, u) = (u-h) (w— i+;^«_2W— 2+ . . . +h,) (mod z^/") 



where h, /?«_2,- • - , ^o are polynomials of degree X — 1 in the element 

 z^''*'. Here X may be taken as great as we will. Assume for the moment 

 that we have n = \ and take X = 2?w+1. The function /(g, h), regarded 

 as a series in powers of s, is then divisible by s^*""*"^ and the function 

 f'u (2, /î), as a consequence of the identity (2), is divisible by the power z*" 

 at most. Substituting u = t-\-h in (1) we obtain an equation 



13. /''+g«_i/«-'+ . . . +e,t+e, = 



in which the coefficients ^„_i, . . . , e^ are series in powers of z involving 

 no negative exponents. In particular we have ei =fi{z,h), eo = f{z, h). 

 The series ei is precisely divisible by a power z*"' where mi<m while e^ 

 is divisible by the power z^*""^^. 



By well known means it may now be shewn] that the equation (13) 

 is satisfied by a convergent series of the form 



14. / = aiz'«+^+a2Z*"+2^_ 



To this series adding the polynomial h we obtain a series u = Pi which 

 satisfies the equation f(z,u)=0. We have for the moment assumed 

 H=l. If this is not the case we substitute f = z^'''' and the preceding 

 argument then proves the existence of a series u = Pi in powers of f 

 which satisfies the equation. We can then split off a linear factor u — Pi 

 from /(z, u) and successive applications of the process splits /(z, u) into 

 a product 

 15. fiz,u)^(u-P,)...(u-PJ 



where, as we know, the series Pi, . . . , P„ group themselves in cycles. 



