The History of Mathematics: A Brief Course

444 15. MODERN ALGEBRA

1.6. Ruffini. As it turned out, Gauss had no need to publish his own research

on the quintic equation. In the very year in which he wrote his dissertation, the

first claim of a proof that it is impossible to find a formula for solving all quintic

equations by algebraic operations was made by the Italian physician Paolo Ruffini

(1765-1822). Ruffini's proof was based on Lagrange's count of the number of values

a function can assume when its variables are permuted.^5 The principles of such

a proof were gradually coming into focus. Waring's proof that every symmetric

function of the roots of a polynomial is a function of its coefficients was an important

step, as was the idea of counting the number of different values a rational function

of the roots can assume. To get the general proof, it was necessary to show that

the root extractions performed in the course of a hypothetical solution would also

be rational functions of the roots. That this is the case for quadratic and cubic

equations is not difficult to see, since the quadratic formula for solving x^2 — (ri +

r 2 )x + ÃéÃ2 = 0 involves taking only one square root:

V(ri + r 2 )^2 - 4rir 2 = \/(ð -r 2 )^2.

Similarly, the Cardano formula for solving y^3 + (r\r 2 +r 2 r 3 +r 3 ri )y = r\r 2 r 3 , where

T\ + r 2 + r 3 = 0, involves taking

/ (nr 2 + r 2 r 3 + r 3 n)^3 (r,r 2 r 3 )^2 [-If, WO 2 , C , „ 2Ë

V 2º + 4 = V LOG [{ri ~

Ã2)(2Ã?

+

5ÃÉÃ2

+ 2ô2ç '

followed by extraction of the cube roots of the two numbers

n+Ljr n2

2 ) and —-=(ôë +u^2 r 2 )

where ù = —1/2 + À\/3/2 is a complex cube root of 1. These radicals are conse-

quently rational (but not symmetric) functions of the roots.

1.7. Cauchy. Although Ruffini's proof was not generally accepted by his contem-

poraries, it was endorsed many years later by Augustin-Louis Cauchy (1789-1856).

In 1812 Cauchy wrote a paper "Essai sur les fonctions symetriques" in which he

proved the crucial fact that a function of ç variables that assumes fewer values than

the largest prime number less than ç when the variables are permuted, actually as-

sumes at most two values. In 1815 he published this result.

Cauchy gave credit to Lagrange, Alexandre Theophile Vandermonde (1735-

1796), and Ruffini for earlier work in this area. Vandermonde, in particular, exhib-

ited the Vandermonde determinant

det

1 x\ x\

1 X 2 X 2 • * • J?2

X^2 · · • xn~^1

= -(×é - X 2 ){XI -× 3 )··· (Xl - ×ç)(×2 - Xa) · • • (X2 -£„)··• (xn-l ~ Xn) ,

which assumes only two values, since interchanging two variables permutes the rows
of the determinant and hence reverses the sign of the determinant.

(^5) An exposition of Ruffini's proof, clothed in modern terminology that Ruffini would not have
recognized, can be found in the paper of Ayoub (1980).