Section III., 1914 [1151 Trans. R.S.C. 



To determine by rational operations -whether an algebraic curve 

 is or is not reducible. 



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



(Read May 27, 1914.) 



We suppose the variables (s, u) to be connected by an algebraic 

 equation 



1. f(z,ll) = U" +/„-l//"- 1 + +/o = 



which equation we may assume to offer no repeated factor. Consider 

 the product of two rational functions. 



A(z,u)=A n -iU n - 1 + +A , 



B(z,u)=B n ^u"- l + +B . 



This product we can write in the form 



2. A (z, u) B (z, u) =f(z, u) g (z, u) + Cn-iU»- l + + C 



where g (s, u) has the form - 



g(z,u)=gn-2 ""~ 2 + +go • 



Identifying coefficients of like powers of u on the two sides of ' 

 (2) we obtain 



3. 2 Ar-tBt = \ fr-tgt , r = n, «+1 ..,2w-2, 



t^r-n+\ l=r-n 



4. 2 Ar-tB t = 2 fr-tgt + C\- , r = o, 1 n-1. 



t=0 t=0 



If now we suppose A (z, u) to represent the general rational function 

 conditioned by a certain set of orders of coincidence (t)' for finite 

 values of the variable z, the necessary and sufficient condition that the 

 tunction B(z,u) may have for finite values of z orders of coincidence 

 which are complementary adjoint to those furnished by (t)' is that 

 the coefficient C n -\ in (2) be integral*. In particular if A (z, u) be the 

 general rational^ function of integral algebraic character we shall 

 have in jB(z,w)*the representation of the general rational function 

 which is adjoint for finite values of z on subjecting its coefficients 



B n -\ B to just those conditions which suffice to make C n -\ 



integral. Now Hensel has shown** how to construct the general rational 

 function of integral algebraic character by a process involving rational 

 operations only. We shall then assume that A{z,u) is the general 

 rational function of integral algebraic character. 



* On the foundations of the theory of algebraic functions of one variable. 

 Phil. Trans. Roy. Soc, Ser. A, vol. 212, pp. 345, 347. 



** Théorie des fonctions algébriques d'une variable. Acta Mathematica, vol. 18, 

 pp. 247-317. See. also H. F. Baker, Abelian Functions, pp. 105-110. 



