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

5.7. Some algebraic tools used by GAUSS 95
was the 3-cycle
� �
αβγ
. (5.16)
γαβ
Thus, given any 3-cycle (5.16), the two p-cycles (5.15) could be formed. Under the hy-
pothesis, these p-cycles left K unaltered, whereby the same was true of their product,
i.e. the 3-cycle (5.16).
In the third and final part of the proof CAUCHY established that if the value of
K was unaltered by all 3-cycles, the function K would either be symmetric or have
two different values. In his proof, analogous to the second part described above, he
decomposed the 3-cycle
� �
αβγ
γαβ
into the product of the two transpositions
� �� �
αβ βγ
βα γβ
which he wrote as (αβ) (βγ). This step of the proof corresponds to proving that the
alternating group An is generated by all 3-cycles.
In the remaining part of the paper, CAUCHY demonstrated for functions of six
arguments, if R < 5 the function would necessarily be symmetric or have two values.
Generally, CAUCHY noted, for n > 4 no functions of n quantities were known which
had less than n values without this number being either 1 or 2. After these two early
papers on the theory of permutations, CAUCHY would let the topic rest for 30 years
being preoccupied with his many other research themes and his teaching. When he
finally returned to the theory of permutations in the 1840s, CAUCHY demonstrated the
following generalization of his 1815 result: That no function of n quantities could take
on less than n values without either being symmetric or taking on exactly 2 values. 99
With his paper, 100 CAUCHY founded the theory of permutations by providing it
with its principal objects: the permutations. He introduced terms and notation which
enabled him to grasp the substitutions as objects abstracted from their action on the
formal values of a function, and he provided an important theorem in this new theory
which he based on an elegant, non-computational proof.
5.7 Some algebraic tools used by GAUSS
GAUSS’ first proof of the Fundamental Theorem of Algebra had, in a central way, de-
pended on geometrical (topological) intuitions. In 1815, GAUSS published a second
proof of the theorem, 101 this time applying algebraic methods. In the process, GAUSS
99 (Dahan, 1980, 281–282).
100 (A.-L. Cauchy, 1815a).
101 (C. F. Gauss, 1815). Eventually, GAUSS would publish two further proofs (one in 1816 and one in
1849) bringing his total to four.