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

174 Chapter 8. A grand theory in spe
For any u > 1, ABEL had, therefore, explicitly demonstrated that pus was a rational
function of the roots z0, . . . , zµ−1.
The irreducible equation for s was Abelian. The ultimate result of ABEL’S studies
of the solubility of equations amounted to a characterization of the irreducible equation
P = 0 which the quantity s satisfied. By arguments founded in C. F. GAUSS’ (1777–
1855) theory of primitive roots, ABEL found that P = 0 had the property of having all
its roots representable as the “orbit” of a rational function (see page 145) whereby the
equation fell into the category studied in the Mémoire sur une classe particulière. 24
Denoting the degree of the irreducible equation P = 0 by ν, ABEL could express its
ν roots in one of the two forms
s or p µ m k1 s m k 1 for 1 ≤ k1 ≤ ν − 1
where m k1 ∈ {2, 3, . . . , µ − 1}. He deduced this from (8.6) described above, since
choosing any other root extraction would give an ˆs of the form p µ
θ sθ . Fixing some m, a
sequence could be constructed, possibly renumbering the coefficients p0, . . . , p k1−1,
s1 = p µ
0 sm ,
s2 = p µ
1 sm 1 ,
.
s k1 = p µ
k 1−1 sm k 1−1 .
At some point, the sequence would stabilize because only finitely many different roots
of P = 0 could be listed. Assuming this to have occurred after the k th
1
which point the value could be assumed to be s again, ABEL wrote
s = sk1 = p µ
k1−1sm k1−1 = smk k1−1 1
∏ p
u=0
µmu
k1−(u+1) .
Dividing this equation by s and extracting the µ th root, he obtained the relation
s mk1 −1
µ
k1−1 ∏ p
u=0
mu
k = 1.
1−u−1
iteration, at
Since the product was a rational function of s by the previous result, ABEL concluded
that the exponent of s would have to be integral
24 (N. H. Abel, 1829c)
m k 1 − 1
µ
= integer,
or m k 1 ≡ 1 (mod µ) .