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

148 Chapter 7. Particular classes of solvable equations
algebraically solvable if y0 was considered known. Although the equation of degree
m
m−1
∏ (z − yu) = 0 (7.12)
u=0
giving the coefficients of (7.11) would generally not be algebraically solvable, ABEL
next proved that equation (7.12) ‘inherited’ the property of commutative rational de-
pendence among its roots, which φ (x) = 0 possessed. Thus, a ‘descent’ down a string
of equations was made possible.
ABEL’S proof of this ‘inheritance’, the commutative rational dependence among
the roots of (7.12), ran as follows. The hypothesis was that all the roots were given
rationally in a single root, i.e.
xu = θu (x0) for 0 ≤ u ≤ m − 1. (7.13)
The expression for yu which in the previous argument was given by (7.3),
�
yu = f xu, θ (xu) , . . . , θ n−1 �
(xu) for 0 ≤ u ≤ m − 1,
under the current hypothesis became
�
y1 = f θ1 (x0) , θ (θ1 (x0)) , . . . , θ n−1 �
(θ1 (x0)) .
Combining this with the hypothesis of commutativity of the functions θ and θ1, ABEL
found
y1 = f
�
�
θ1 (x0) , θ1 (θ (x0)) , . . . , θ1 θ n−1 ��
(x0) .
Therefore, y1 was a rational and symmetric function of the sequence of roots (7.13) and
could therefore be expressed rationally in y0 and known quantities. Obviously, ABEL
could carry out the same argument for any other y2, . . . , ym−1. When he let λ (y0) and
λ1 (y0) denote any two among the quantities y0, . . . , ym−1, he found that, without loss
of generality,
y1 = λ (y0) = f
y2 = λ1 (y0) = f
�
θ1 (x0) , θ (θ1 (x0)) , . . . , θ n−1 �
(θ1 (x0)) and
�
θ2 (x0) , θ (θ2 (x0)) , . . . , , θ n−1 �
(θ2 (x0)) .
Inserting θ2 (x) for x0 in λ (y0), which transformed y0 into y2, ABEL obtained 10
λλ1 (y0) = λ (y2) = f
while inserting θ1 (x) for x0 in λ1 (y0) produced
λ1λ (y0) = λ1 (y1) = f
�
θ1θ2 (x0) , θθ1θ2 (x0) , . . . , θ n−1 �
θ1θ2 (x0)
�
θ2θ1 (x0) , θθ2θ1 (x0) , . . . , θ n−1 �
θ2θ1 (x0)
10 Here I deviate from my usual notation by writing the composition of functions in multiplicative
mode, i.e. θ1θ2 (x0) instead of θ1 (θ2 (x0)).
,
.