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

166 Chapter 8. A grand theory in spe
could occur in the solution of a solvable equation. This characterization had been one
of the points of objection to his impossibility proof of 1826 (see section 6.9.1). However,
only the objections raised by E. J. KÜLP (⋆1801) were known to ABEL and ABEL did
not react directly to them in the notebook. The characterization which ABEL presented
in the notebook was only a limited version of the one found in the impossibility proof.
In the notebook, ABEL described the radicals from the outer-most one inward in the
following form
1
µ 1
2
µ 1
y = P0 + P1 · R1
+ P2 · R1
+ · · · + Pµ 1−1 · R1
, (8.1)
in which P0, . . . , Pµ 1−1 and R1 were rational expressions in known quantities and the
1 1
µ 2 µ 3
other radicals R2
, R3
, . . . . In relation to the route he had taken in the impossibility
proof, he abandoned the concept of degree of algebraic expressions and imposed only
the hierarchy from the concept of order which counted the number of nested root
extractions of prime degree.
ABEL introduced three notational concepts which he used throughout the prelimi-
nary part of the manuscript to simplify his notation:
1. He chose to denote algebraic expressions by writing their order as subscripts, for
µ 1 −1
µ 1
instance writing Am for an algebraic expression A of order m.
2. With y being of the form (8.1) and φ (y) = 0 an equation satisfied by y, ABEL
chose to write the equation as φ (y, m) = 0 if all the coefficients of φ (y) were
algebraic expressions of order m. Furthermore, he denoted the degree of the
equation by δφ (y, m).
3. Most importantly, he introduced a symbol ∏ Am for the product of all values of
Am obtained from attributing to the outermost radical in Am, R 1 µ , all its possible
values, R 1 µ , ωR 1 µ , . . . , ω µ−1 R 1 µ (ω a µ th root of unity). Thus, if
Am =
the new symbol denoted the expression
∏ Am =
µ−1
∑ pkR k=0
k µ ,
�
µ−1 µ−1
∏ ∑ pkω u=0 k=0
uk R k �
µ .
Using these concepts and a number of immediate consequences derived from them,
ABEL constructed and characterized the irreducible equation associated with a given
algebraic expression.