RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

5.1. Algebraic solubility before LAGRANGE 63
The quotation illustrates how EULER’S conjecture amounted to the algebraic solu-
bility of all polynomial equations. Returning to the problem, EULER sought to provide
further evidence for his conjecture. 21
EULER was led to a related problem concerning the multiplicity of values of rad-
icals. By calculating the number of values of the multi-valued function consisting of
n − 1 radicals
n√ A + n √ B + n√ C + n√ D + . . . ,
EULER found that the function had n n−1 essentially different values, which apparently
contradicted the fact that the equation of degree n should only have n roots. In a paper
written in 1759, EULER refined his hypothesis of 1732 and conjectured that the roots
of the resolvent A, B, C, D were dependent. EULER’S new conjecture was that the root
would be expressible in the form
x = ω + A n√ v + B n√ v 2 + C n√ v 3 + · · · + D n√ v n−1 ,
where the coefficients ω, A, B, C, . . . , D were rational functions of the coefficients, and
the n − 1 other roots would be obtained by attributing to n√ v the n − 1 other values
a n√ v, b n√ v, c n√ v . . . where a, b, c were the different n th roots of unity. 22 As will be illus-
trated in chapter 7.1.2, N. H. ABEL (1802–1829) used a similar kind of argument.
20 “8. Ex his etiamsi tribus tantum casibus tamen non sine sufficienti ratione mihi concludere videor
superiorum quoque aequationum dari huiusmodi aequationes resolventes. Sic proposita aequatione
coniicio dari aequationem ordinis quarti
cuius radices si sint A, B, C et D, fore
Et generatim aequationis
x 5 = ax 3 + bx 2 + cx + d
z 4 = αz 3 − βz 2 + γz − δ,
x = 5√ A + 5√ B + 5√ C + 5√ D.
x n = ax n−2 + bx n−3 + cx n−4 + etc.
aequatio resolvens, prout suspicor, erit huius formae
z n−1 = αz n−2 − βz n−3 + γz n−4 − etc.
cuius cognitis radicibus omnibus numero n − 1, quae sint A, B, C, D etc., erit
x = n√ A + n√ B + n√ C + n√ D + etc.
Haec igitur coniectura si esset veritati consentanea atque si aequationes resolventes possent determinari,
cuiusque aequationis in promtu foret radices assignare; perpetuo enim pervenitur ad
aequationem ordine inferiorem hocque modo progrediendo tandem vera aequationis propositae
radix innotescet.” (L. Euler, 1732b, 7–8); for a German translation, see (L. Euler, 1788–1791, vol. 3,
9–10).
21 (F. Rudio, 1921, ix–x).
22 (ibid., x–xi).