446 15. MODERN ALGEBRA1.8. Abel. Cauchy's work had a profound influence on two young geniuses whose
lives were destined to be very short. The first of these, the Norwegian mathe-
matician Niels Henrik Abel (1802-1829), believed in 1821 that he had succeeded
in solving the quintic equation. He sent his solution to the Danish mathematician
Ferdinand Degen (1766-1825), who asked him to provide a worked-out example of
a quintic equation that could be solved by Abel's method. While working through
the details of an example, Abel realized his mistake. In 1824 he constructed an
argument to show that such a solution was impossible and had the proof published
privately. A formal version was published in the Journal fur die reine und ange-
wandte Mathematik in 1826. Abel was aware of Ruffini's work, and mentioned it
in his argument. He attempted to fill in the gap in Ruffini's work with a proof that
the intermediate radicals in any supposed solution by formula can be expressed as
rational functions of the roots.
Abel's idea was that if some finite sequence of rational operations and root
extractions applied to the coefficients produces a root of the equation
xb - ax^4 + bx^3 - cx^2 + dx - e = 0,
the final result must be expressible in the form× = p + #™ +p 2 R™ ë rPm-lR^ ,
where p, p2,--, Pm-ii and R are also formed by rational operations and root
extractions applied to the coefficients, m is a prime number,^7 and R)lm is not
expressible as a rational function of the coefficients a, b, c, d, e, p, p 2 ,..., pm_i.^8
By straightforward reasoning on a system of linear equations for the coefficients pj,
he was able to show that R is a symmetric function of the roots, and hence that
R]/'n must assume exactly m different values as the roots are permuted. Moreover,
since there are 5! permutations of the roots and m is a prime, it followed that
m = 2 or m = 5, the case m = 3 having been ruled out by Cauchy. The hypothesis
that m = 5 led to an equation in which the left-hand side assumed only five values
while the right-hand side assumed 120 values as the roots were permuted. Then the
hypothesis m = 2 led to a similar equation in which one side assumed 120 values
and the other only 10. Abel concluded that the hypothesis that there exists an
algorithm for solving the equation was incorrect.
The standard version of the history of mathematics credits Abel with being
"the" person who proved the impossibility of solving the quintic equation. But
according to Ayoub (1980, p. 274), in 1832 the Prague Scientific Society declared
the proofs of Ruffini and Abel unsatisfactory and offered a prize for a correct proof.
The question was investigated by William Rowan Hamilton in a report to the Royal
Society in 1836 and published in the Transactions of the Royal Irish Academy in
- Hamilton's report was so heavily laden with subscripts and superscripts bear-
ing primes that only the most dedicated reader would attempt to understand it,
although Felix Klein was later (1884) to describe it as being "as lucid as it is volu-
minous." The proof was described by the American number theorist and historian
(^7) Extracting any root is tantamount to the sequential extraction of prime roots. Hence every root
extraction in the hypothetical process of solving the equation can be assumed to be the extraction
of a prime root.
(^8) Abel incorporated the apparently missing coefficient pi into R here, since he saw no loss of
generality in doing so. A decade later, Hamilton pointed out that doing so might increase the
index of the root that needed to be extracted, since p\ might itself require the extraction of an
mth root.