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

102 Chapter 6. Algebraic insolubility of the quintic
troduced algebraic expressions as objects, ABEL derived a standard form for these ob-
jects. Applying it to algebraic expressions which satisfied a given equation, he found
that these could always be given a form in which all occurring components depended
rationally on the roots of the equation.
In his effort to obtain a classification of algebraic expressions, ABEL introduced
a hierarchy based on the concepts of order and degree. These concepts introduced a
structure in the class of algebraic expressions allowing ordering and induction to be
carried out.
In dealing with the proof which ABEL gave of his auxiliary theorem, we are in-
troduced to two other concepts which are even more fundamental to his theory of
algebraic solubility. These are the Euclidean division algorithm and the concept of ir-
reducibility. In section 6.3.3, the proof is presented in quite some detail to demonstrate
how ABEL made use of these concepts. They were to become even more important in
his unpublished general theory of solubility (see chapter 8).
6.3.1 Orders and degrees
ABEL’S classification of algebraic functions (expressions) was hierarchic; his means to
obtain structure were the two concepts of order and degree. The order was introduced
to capture the depth of nested root extractions, whereas the degree kept track of root
extractions at the same level by imposing a finer structure. ABEL defined rational
expressions to be of order 0, and the order concept was thereafter defined inductively.
Thus, if f was a rational function of expressions of order µ − 1 and root extractions
of prime degree of such expressions, f would be an algebraic expression of order µ.
With this idea, ABEL obtained the following standard form of algebraic expressions of
order µ:
f (g1, . . . , gk; p√ 1 r1, . . . , pm √ rm) , (6.1)
where f was a rational expression, the expressions g1, . . . , g k and r1, . . . , rm were alge-
braic expressions of order µ − 1, and p1, . . . , pm were primes.
Thus, as indicated, ABEL’S concept of order counted the number of nested root
extractions of prime degree. For instance, if R was a rational function (i.e. of order 0),
√ R was of order 1, 3 �√ R of order 2, and similarly 3 �√ R + √ R was of order 2. Also
4√ R was of order 2, since it would have to be decomposed as two nested square roots,
�√ R.
Within each order, ABEL described another hierarchy controlled by the concept
of degree. While the order served to denote the number of nested root extractions
of prime degree, ABEL’S concept of the degree of an algebraic expression counted
the number of co-ordinate root extractions at the top level. Thus in (6.1), it was the
minimal value of m which would suffice to write the expression in this form. In table
6.1, I have illustrated the concepts by listing the orders and degrees of one of the