by means of Radicals. 387 



is a six-valued function. And it is in virtue of tins exception 

 that a cubic or a quartic equation is aolvible by radicals. But I 

 assume that lor // -- 4 (he Lemma is true without exception. 



The course of demonstration would be something as follows : — 

 Imagine, if possible, the root of an equation expressed, by means 

 of radicals, in terms of the coefficients ; the expression cannot 

 contain any radical such as \/X, p > 2, where X is a one-valued 

 (or rational) function of the coefficients, not a perfect /;th power, 

 for the reason that, expressing the coefficients in terms of the 

 roots, such function \/X * s n °t a rational function of the roots ; 

 if it were so, by lemma I. it would be a one-valued (that is, a 

 symmetrical) function of the roots; consequently a rational func- 

 tion of the coefficients, or X expressed in terms of the coefficients, 

 would be a perfect /;th power. 



The expression may however contain a radical \/Xj X a one- 

 valued (or rational) function of the coefficients, not a perfect 

 square ; viz. X may be any square function multiplied into that 

 function of the coefficients which is equal to the product of the 

 squared differences of the roots, or, say, multiplied into the dis- 

 criminant X = Q 2 V, or \/X = Q\/v- 



We have next to consider whether the expression can contain 

 any radical v^X, where X, not being a rational function of the 

 coefficients, is a function expressible by radicals. But the fore- 

 going reasoning shows that if this be so, X cannot contain any 

 radical other than the radical \/Q 2 V or Q\/v> as above ; that 

 is, X must be =P + Q\/v> where P and Q are rational functions 

 of the coefficients, and where we may assume that P + QvV is 

 not a perfect jt?th power of a function of the like form P'-|- Q'\/\7. 

 But then, expressing the coefficients in terms of the roots, we 

 have P-r-Q/\/v> a (rational) two-valued function of the roots; 

 and there is no radical vP + Qv/y, which is a rational func- 

 tion of the roots ; for by lemma II., if such radical existed 



we should have vP + t^/v, a (rational) two-valued function of 

 the roots; that is, it would be = P' + Q'\/v> P'andQ' one- valued 

 (symmetrical) functions of the roots, consequently rational 

 functions of the coefficients; or P + Q\/v would be a perfect 



jpth power (P' + Q , v / V) p . 



The conclusion is that for n > 4 there is not (besides the 

 function P + Q-s/v) an y function of the coefficients, expressible 

 by means of radicals, which, when the coefficients are expressed 

 in terms of the roots, will be a rational function of the roots, 

 and consequently no possibility of expressing the roots in terms 

 of the coefficients by means of radicals. 



Cambridge, October 1, 1868. 



2C2 



