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

104 Chapter 6. Algebraic insolubility of the quintic
After denoting by V1, . . . , Vp−1 the values of V by inserting α k p √ R for p√ R in V (α a p th
root of unity), ABEL multiplied numerator and denominator of (6.3) by V1V2 . . . Vp−1.
The denominator thereby became a rational function of r1, . . . , r k “as it is known”. 18
The conclusion can be seen as an application of ABEL’S implicit version of LAGRANGE’S
theorem 1. 19
By this analogous of multiplying the denominator by conjugates, 20 ABEL had shown
that the expression v could be written as a polynomial in p√ R,
v = f
�
r1, . . . , rk, p√ � p−1
R =
∑
u=0
quR u p ,
where R was of order µ − 1 and all the coefficients q0, . . . , qp−1 were functions of order
µ and degree at most m − 1 such that R 1 p could not be expressed rationally in the coeffi-
cients. ABEL also stated that the coefficient q1 could be assumed equal to 1. In this last
step, ABEL’S conclusions concerning the orders and degrees of the other coefficients
were too bold, as W. R. HAMILTON (1805–1865) and L. KÖNIGSBERGER (1837–1921)
in 1839 and 1869, respectively, were to point out (see section 6.9.1). 21 In general, this
step — obtained by dividing each coefficient by q1 — might effect the order of R which
could now be µ. However, as KÖNIGSBERGER also noticed, the mistake was not an
essential one and has no consequences for the rest of the proof (see section 6.9.1).
In ABEL’S version, the standard form of algebraic expressions can be described by
theorem 2.
Theorem 2 Let v be an algebraic expression of order µ and degree m. Then
v = q0 + p 1 n + q2p 2 n + · · · + qn−1p n−1
n , (6.4)
where n is a prime, q0, q2, . . . , qn−1 are algebraic expressions of order µ and degree at most
m − 1, and p is an algebraic expression of order µ [ABEL stated µ − 1, see below] such that
p 1 n cannot be expressed as a rational function of q0, q2, . . . , qn−1. (N. H. Abel, 1826a, 70) ✷
In his modified version, KÖNIGSBERGER only concluded that the algebraic expression
p was of order µ and degree at most m − 1, and that the order of p 1 n was µ.
18 (N. H. Abel, 1826a, 69).
19 The function V can be interpreted as depending upon all the roots of the equation X p = R, i.e.
�√p p
V = V R, α √ R, . . . , αp−1 p √ �
R although only the first argument is actually involved. The values
V0, . . . , Vp−1 are then obtained by transposing the first argument with any other argument, and
the theorem 1 states that the product ∏ p−1
u=0 Vu is a rational function of
p√
R, . . . , αp−1 p √ R and the
coefficients of V.
20 In order to obtain a real denominator of the fraction
a + ib
c + id
its numerator and denominator are both multiplied by c − id.
21 (W. R. Hamilton, 1839; Königsberger, 1869).