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

62 Chapter 5. Towards unsolvable equations
LEONHARD EULER. In his paper (L. Euler, 1732b), read to the St. Petersburg Academy
and published in 1738, EULER gave his solutions to the equations of degree 2, 3, and 4
and demonstrated that they could all be written in the form 16
√ A for the second degree equation,
3√ A + 3 √ B for the third degree equation, and
4√ A + 4 √ B + 4√ C for the fourth degree equation,
(5.4)
where the quantities A, B, C were roots in certain resolvent equations of lower degree
which could be obtained from the original equation. 17 EULER appears to have been
the first to introduce the term “resolvent” and to attribute to it the central position it
was to take in the future research on the solubility of equations. 18
Extending these results, EULER conjectured that the resolvents also existed for the
general equation of the fifth degree — and more generally for any higher degree equation
— and that the roots could be expressed in analogy with (5.4). 19
“Although this emphasizes the three particular cases [of equations of degrees
2, 3, and 4], I, nevertheless, think that one could possibly, not without reason, conclude
that also higher equations would possess similar solving equations. From
the proposed equation
x 5 = ax 3 + bx 2 + cx + d,
I expect to obtain an equation of the fourth degree
z 4 = αz 3 − βz 2 + γz − δ
the roots of which will be A, B, C, and D,
In the general equation
x = 5√ A + 5√ B + 5√ C + 5√ D.
x n = ax n−2 + bx n−3 + cx n−4 + etc.
the resolvent equation will, I suspect, be of the form
z n−1 = αz n−2 − βz n−3 + γz n−4 − etc.,
whose n − 1 known roots will be A, B, C, D, etc.,
z = n√ A + n√ B + n√ C + n√ D + etc.
If this conjecture is valid and if the resolvent equations, which can obviously be
said to have assignable roots, can be determined, I can obtain equations of lower
degrees, and in continuing this process produce the true root of the equation.” 20
16 (ibid., 7).
17 The resolvent equation in the example of the third degree equation is (5.3).
18 (F. Rudio, 1921, ix, footnote 2).
19 According to (Eneström, 1912–1913, 346) already LEIBNIZ seemed conviced that the root of the general
equation of the 5 th degree could be written in the form
x = 5√ A + 5√ B + 5√ C + 5√ D.