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

170 Chapter 8. A grand theory in spe
Construction of the irreducible equation. With the first theorems and the lemmata
described above, ABEL was in a position to give a construction of the irreducible equa-
tion which a given algebraic expression satisfied. More importantly, this construction
allowed him to demonstrate that central properties of this equation could be deduced
from properties of the initially given algebraic expression. ABEL let
am = f ( µm √ ym, µ √
m−1 ym−1, . . . )
denote a given algebraic expression and constructed the irreducible equation ψ (y) = 0
which would have am as a root in the following way.
Since am was to satisfy ψ (y) = 0, it would be necessary that y − am was a factor of
ψ. By the theorem 10, it followed that
φ1 (y, m1) = ∏ (y − am)
would also be a factor. Because y − am was a first degree polynomial and, therefore,
irreducible, it followed that φ1 was also irreducible (by theorem 11). Consequently,
φ1 was an irreducible factor of ψ (y) and the procedure could be repeated yielding a
sequence of irreducible factors
φn (y, mn) = ∏ φn−1 (y, mn−1) ,
in which the radicals of am were sequentially removed by the analogue of multiplying
with the complex conjugate (c.f. section 6.3.2).
ABEL claimed that the sequence of positive integers m1, m2, . . . was decreasing but
gave no explicit argument. However, by J. L. LAGRANGE’S (1736–1813) theorem (sec-
tion 5.2.3) it is not hard to see that ∏ am is a rational function of ym and the inner
radicals involved. Therefore, the order of ∏ am is less than the order of am. Thus,
at a certain point (after, say, u steps) the sequence m1, m2, . . . had to vanish, and an
equation would be obtained in which all the coefficients were rationally known. This
equation was the sought-for ψ (y) = 0,
ψ (y) = φu (y, 0) = ∏ φu−1 (y, mu−1) .
Directly from this construction, ABEL deduced his characterization of the irre-
ducible equation satisfied by a given algebraic expression, laying the foundations for
his further reasoning. He summarized the properties in the following four points: 15
Proposition 1 The following four results link properties of the irreducible equation ψ (y) = 0
satisfied by a given algebraic expression am to properties of the expression itself:
1. The degree of ψ is the product of certain exponents of root extractions occurring in am.
Among these exponents, the one of the outer-most root extractions is always present.
15 (N. H. Abel, [1828] 1839, 232–233)