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

8.4. Further ideas on the theory of equations 179
radicals. In this way, the insolubility of fifth degree equations of the standard form 28
x 5 + ax + b = 0 was demonstrated directly: If the equation had been solvable, ABEL
possessed a solution formula, which he saw was not powerful enough to give the
solution of arbitrary equations.
In a letter to HOLMBOE from the same year, the result on the form of roots was
given another twist.
“Concerning equations of the 5 th degree I have found that whenever such an
equation can be solved algebraically, the root must have the following form:
x = A + 5√ R + 5√ R ′ + 5√ R ′′ + 5√ R ′′′
where R, R ′ , R ′′ , R ′′′ are the 4 roots of an equation of the 4 th degree and have the
property that they can be expressed with help of only square roots. — It has been
a difficult task for me with respect to expressions and notation.” 29
In this form, the statement is a refined version of EULER’S “conjecture” that the
solution of the fifth degree equation should be of the form
A + 5√ R + 5√ R ′ + 5√ R ′′ + 5√ R ′′′ (8.9)
where R, R ′ , R ′′ , R ′′′ were solutions to an equation of the fourth degree (see section
5.1).
wo
�
a = m + n 1 + e2 +
�
a1 = m − n
�
a2 = m + n
�
a3 = m − n 1 + e2 +
�
�
h 1 + e2 + �
1 + e2 �
,
1 + e2 � �
+ h 1 + e2 − �
1 + e2 �
,
1 + e2 � �
− h 1 + e2 + �
1 + e2 �
,
� �
h 1 + e2 − �
1 + e2 �
,
A = K + K ′ a + K ′′ a2 + K ′′′ aa2,
A1 = K + K ′ a1 + K ′′ a3 + K ′′′ a1a3,
A2 = K + K ′ a2 + K ′′ a + K ′′′ aa2,
A3 = K + K ′ a3 + K ′′ a1 + K ′′′ a1a3.
Die Grössen c, b, e, m, n, K, K ′ , K ′′ , K ′′′ sind alle rationale Zahlen.
Auf diese Weise lässt sich aber die Gleichung x 5 + ax + b = 0 nicht auflösen, so lange a und b
beliebige Grössen sind.” (Abel→Crelle, Freyberg, 1826/03/14. N. H. Abel, 1902a, 21–22).
28 If formulated in positive way, the researches of JERRARD (see section 6.9.1) demonstrated that every
fifth degree equation could be transformed to this normal trinomial form. (W. R. Hamilton, 1839,
251)
29 “Med Hensyn til Ligninger af 5th Grad har jeg faaet at naar en saadan Ligning lader sig løse algebraisk
maa Roden have følgende Form:
x = A + 5√ R + 5√ R ′ + 5√ R ′′ + 5√ R ′′′
hvor R, R ′ , R ′′ , R ′′′ ere de 4 Rødder af en Ligning af 4de Grad, og som lade sig udtrykke blot ved
Hjelp af Qvadratrødder. — Det har været mig en vanskelig Opgave med Hensyn til Udtryk og
Tegn.” (Abel→Holmboe, Paris, 1826/10/24. N. H. Abel, 1902a, 45).