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

178 Chapter 8. A grand theory in spe
where a0, . . . , aν−1 were roots of an irreducible Abelian equation of degree ν and the
coefficients A i were rational expressions in s. The root z0 of the initial equation was in
turn given from the sequence s
1
µ
0
, . . . , s
z0 = p0 +
1
µ
ν−1
ν−1
∑
u=0 k=0
by a relationship of the form
ν−1
∑
m
φu (sk) · s
u
µ
k ,
where φ0, . . . , φν−1 were rational functions. In a letter to CRELLE dated 1826, ABEL
had announced a result for equations of the fifth degree which was a particular case
of the above.
“When an equation of the fifth degree, whose coefficients are rational numbers,
is algebraically solvable, one can always give its roots the following form:
x = c + A · a 1 5 · a 2 5
1 · a 4 5
2 · a 3 5
3 + A1 · a 1 5
1 · a 2 5
2 · a 4 5
3 · a 3 5
where
�
a = m + n 1 + e2 +
�
a1 = m − n
�
a2 = m + n
+ A2 · a 1 5
2 · a 2 5
3 · a 4 5 · a 3 5
1 + A3 · a 1 5
3 · a 2 5 · a 4 5
1 · a 3 5
2
�
h
�
1 + e2 + h
�
1 + e2 − h
�
�
a3 = m − n 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.
h
�
1 + e2 + �
1 + e2 �
,
�
1 + e2 − �
1 + e2 �
,
�
1 + e2 + �
1 + e2 �
,
�
1 + e2 − �
1 + e2 �
,
The quantities c, b [h], e, m, n, K, K ′ , K ′′ , K ′′′ are all rational numbers.
In this way, however, the equation x 5 + ax + b = 0 cannot be solved as long as
a and b are arbitrary quantities.” 27
Probably from his realization that all quantities involved in the solution are ratio-
nals, square roots of rationals, or fifth roots of rationals, ABEL concluded that there
were values of a and b for which the equation x 5 + ax + b = 0 could not be solvable by
27 “Wenn eine Gleichung des fünften Grades, deren Coëfficienten rationale Zahlen sind, algebraisch
auflösbar ist, so kann man immer den Wurzeln folgende Gestalt geben:
x = c + A · a 1 5 · a 2 5
1 · a 4 5
2 · a 3 5
3 + A1 · a 1 5
1 · a 2 5
2 · a 4 5
3 · a 3 5
+ A2 · a 1 5
2 · a 2 5
3 · a 4 5 · a 3 5
1 + A3 · a 1 5
3 · a 2 5 · a 4 5
1 · a 3 5
2