[FIELDS] THE EXISTENCE THEOREM 89 



coefficients already determined in these series and in the series 

 qo, g\, . . . , qk-i- 



By repetition of the process employed above suppose that we have 

 determined the constant terms and the coefficients of z, z^, . . ., ^~^ in 

 the series Qo, qi, . . . , qk-i, Qo, Qi, ■ ■ ■ , Qn-k-i in accord with the identi- 

 ties (5) and so that the series Sg, Si, . . . , Sk-i in (6) are all divisible 

 by z^. Equate to in succession the coefficients of z^ in the expressions 

 for So, Si, . . . , Sfi-i given in (6) and we successively determine the 

 coefficients of z^ in the series g^, qi, . . . , q^-i in terms of the coefficients 

 of lower powers of s in these series and in the series Qo, Qi, ■ ■ ■ , Q„-k-i- 

 The identities (5) thereafter determine the coefficients of z^ in the series 

 Qoy Qh ■ ■ ■ ! Qn-k-i iri terms of the coefficients already determined in 

 these series and in the series qg, q\, . . . , qk-i- By induction therefore 

 we conclude that however great X may be we can determine the constant 

 terms and the coefficients of s, 2^, . . ., ^~^ in the series g^, qi, . . . , 

 Qk-iy Qoj Qii ■ • • ) Qn-k-i in such order that each one is expressed ration- 

 ally in terms of those already determined, their determination being 

 made in accord with the identities (5) and in such manner that the ex- 

 pressions So, S\, . . . , Sfi-i in (6) are all divisible by s^. That the con- 

 stant terms and the coefficients have been so determined we shall indicate 

 by writing 



7. Kz, ti)=(tc'+g,_,u'-'+ . . . +çJ(M'-H(2„-fe-i^^''-'-'+ . . . +Qo) 



(mods^). 



If we do not exclude fractional exponents from the coefficients 

 fn-i, . . . , /o in (1) but suppose these coefficients to be series arranged 

 according to integral powers of the element f = 3^^" we can replace 2 by f 

 in the preceding argument. On assuming that /^ is the last of the co- 

 efficients which is not divisible by f we arrive at a factorisation of the 

 form 



8. f{z, «)=(«'+ ç.-i«'-'+ . . . +ço) («""'+ Q„-k-it'"-'-'+- ■ • +a) 



(mod s^/" ) 

 where qg, . . . , qk-i, Qo, ■ ■ •> Qn-k-i are series in powers of f = 2^'^" in 

 which the constant terms and the coefficients of z^^", ..., z^^'^^^" are 

 determined, the constant terms in g<,) • • • > Qk-i being assumed to be 0. 



In what precedes some but not all of the roots of the equation/(o, u) = 

 are equal to 0. The general case where the roots of this equation 

 are not all equal can be immediately reduced to the case just handled. 

 For on writing u=v-{-p, where p is one of the roots of the equation 

 f(o, u) =0, we obtain an equation F(o,v) =0 some but not all of whose 

 roots have the value 0. By what we have seen above then we can efïect a 



