Section III, 1920 I87] Trans. R.S.C. 



Algebraic Proof of the Existence Theorem for the Branches of an Algebraic 

 Function of One Variable. 



By Professor J. C. Fields, F.R.S. 



(Read May Meeting, 1920). 



The proof of the Existence Theorem presented in this paper is one 

 which the writer has given to his students for many years past in a 

 course of lectures on the algebraic functions. The method is almost 

 obvious and its simplicity will perhaps commend itself to those who have 

 occasion to give an introductory course on the theory of the algebraic 

 functions. 



Consider an equation 



1. /(Z,«)=W«+/„_1 «»-!+... +/o =0 



in which the coefficients fn-i, ■ ■ - , fa ^re series in powers of z with 

 integral exponents. We shall suppose that no negative exponents pre- 

 sent themselves. We shall assume that n is greater than 1 and we shall 

 also assume that/(s, w) contains no multiple factor. 



On applying to f(z,u) and fu (s, «), regarded as polynomials in u, 

 the process for finding the greatest common divisor we determine two 

 functions Q{z,u) and R{z,ti), polynomials in u of degrees n—2 and 

 n — 1 respectively with coefficients which are power-series in z, such that 



2. Q(z, u) f{z, u) +Rxz, u) fl (z, n) = s- ('(s)) 



where w is or a positive integer and where in the power-series ((s)) 

 the constant term is different from 0. Furthermore it is readily seen 

 that we may assume not only that Q{z,ti) and R{z, u) are integral with 

 regard to the element z but also that neither of them is divisible by z. 

 For cancellation of the powers of u on the lefthand side of (2) cannot 

 take place if one only of the two expressions here in question is divisible 

 by z. The exponent m is then a perfectly definite number. If on sub- 

 stituting for u in /(z, u) a power-series P{z) we obtain a power-series in z 

 in which the lowest exponent is greater than m it is evident from the 

 identity (2) that the result of substituting P{z) for u in fl (z, u) will be 

 a power-series in z in which the lowest exponent is equal to or less than m. 

 Let us now first assume that the coefficient fo in /(z, u) is divisible 

 by z but that not all of the n coefficients /„_i , . . . ,fo are so divisible. 

 Suppose /;fe to be the last of these coefficients which is not divisible by z. 

 We shall attempt to divide /(z, u) by a polynomial w*-|-gfe-iw*~^+ • • • +?o 

 in which the coefficients q^-i, . . . , Qo are power-series in z as yet 



