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

146 Chapter 7. Particular classes of solvable equations
Obviously, the form of the right hand side shows that this expression was a rational
function of x. ABEL stated that it was unaffected by substituting θ m (x) for x and
considered it so obvious that he did not provide the details. 5 Thus, the expression
was a rational function of the coefficients of φ (x) = 0; ABEL denoted this function by
a k,
µ√
vk = ak ( µ√ v1) k .
v1
ABEL stated the conclusion of this investigation by giving an algebraic formula for
the root x, 6
x = 1
�
−A +
µ
µ√ v1 + a2
(
v1
µ√ v1) 2 + a3
(
v1
µ√ v1) 3 + · · · + aµ−1
(
v1
µ√ v1) µ−1
�
. (7.10)
All the other roots were contained in this formula by giving µ√ v1 its µ different values
α k µ √ v1. ABEL expressed the implications for solubility in two theorems capturing the
essence of this research. If the set of roots fell into one “orbit” of the rational expres-
sion, θ, ABEL found the equation to be solvable by radicals:
Theorem 6 “If the roots of an algebraic equation can be represented by:
x, θx, θ 2 x, . . . θ µ−1 x,
where θ µ x = x and θx denotes a rational function of x and known quantities, this equation
will always be algebraically solvable.” 7
Applying this result to the particular case of irreducible equations of prime degree,
which always had only one chain, ABEL found that such equations were algebraically
solvable:
“If two roots of an irreducible equation, of which the degree is a prime number,
have such a relation that one can express the one rationally in the other, this
equation will be algebraically solvable.” 8
Subsequently, ABEL refined the hypothesis that all the roots could be expressed as
iterations of a rational function. That hypothesis had ensured algebraic solubility of
the equation, but the same conclusion could also be established for a broader class of
equations. Under the general assumption that every root of an equation, χ (x) = 0,
5 The details can easily be provided by inserting into the right hand side and rearranging terms.
6 (N. H. Abel, 1829c, 142).
7 “Si les racines d’une équation algébrique peuvent être représentées par:
x, θx, θ 2 x, . . . θ µ−1 x,
où θ µ x = x et θx désigne une fonction rationelle de x et de quantités connues, cette équation sera
toujours résoluble algébriquement.” (ibid., 142–143).
8 “Si deux racines d’une équation irréductible, dont le degré est un nombre premier, sont dans un
tel rapport, qu’on puisse exprimer l’une rationnellement par l’autre, cette équation sera résoluble
algébriquement.” (ibid., 143).