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

8.3. Refocusing on the equation 171
2. The exponent of the outer-most root extraction divides the degree of ψ [actually contained
in 1, above].
3. If ψ can be algebraically satisfied, it is also algebraically solvable. All its roots are ob-
tained by attributing to the root extractions y
1
µu
mu
all their possible values.
4. If the degree of ψ is µ, the expression am may have µ, and no more than µ, values. ✷
ABEL’S deduction of these properties was straightforward from considerations on
the exponents of involved root extractions and the construction described. A formal
consideration of the uniqueness of the irreducible equation constructed was not car-
ried out, but must have seemed obvious to ABEL.
8.3 Refocusing on the equation
The first theorems and the construction of the irreducible equation connected to a
given algebraic expression are fascinating pieces of mathematics revealing traces of
ABEL’S profound ideas. Whereas the presentation of these fundamental results was
lucid — and basically acceptable to present day mathematicians — ABEL’S following
investigations in the notebook took another form. As he progressed farther from the
well established results founded in the theory of LAGRANGE, his explanatory remarks
and general narrative became ever more sparse until they finally ceased altogether.
However, ABEL’S notebook is the only source illustrating how he planned to proceed,
and I will try to reconstruct the central result of these investigations, which was never
presented in a form intended for publication.
Because ABEL’S argument, from this point onward, consists of little but equations,
I have reconstructed how he could, with his tools and methods, have argued. In lim-
iting myself to ABEL’S argument for the reduction of the general problem to Abelian
equations, I remain close to the sources. ABEL’S unfinished manuscript inspired math-
ematicians of the nineteenth century — such as C. J. MALMSTEN (1814–1886), SYLOW,
and L. KRONECKER (1823–1891) — to elaborate and extend the investigation; 16 recently,
L GÅRDING and C. SKAU have taken up the problem anew. 17
SYLOW has speculated that ABEL recognized the insufficiency of his description
of algebraic expressions. In response to his realization, ABEL should, according to
SYLOW, have abandoned his attempt at presenting a manuscript ready for printing
and instead recorded his further findings in the order and form in which he came to
them. 18 As also noted by SYLOW, this change in style of presentation was not uncom-
mon. In his notebooks, ABEL frequently started out writing coherent manuscripts,
16 (Malmsten, 1847), (Sylow, 1861), (L. Sylow, 1902, 18–22), and (Kronecker, 1856).
17 (Gårding, 1992) and (Gårding and Skau, 1994).
18 (L. Sylow, 1902, 19).