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

6.3. Classification of algebraic expressions 103
Expression
�
�
Order Degree
3
�
3
R + � Q 3 + R 2 + 3
R − � Q 3 + R 2 2 2
R + � Q 3 + R 2 2 1
R + � Q 3 + R 2 1 1
Q 3 + R 2 0 0
Table 6.1: The order and degree of some expressions in CARDANO’S solution to the
general cubic x 3 + a2x 2 + a1x + a0 = 0. R and Q are assumed to be certain rational
functions of the given quantities, here the coefficients a0, a1, a2.
G. CARDANO (1501–1576) solutions to the general cubic. Any rationally related root
extractions were, ABEL said, to be combined and any algebraic expressions of order µ
and degree 0 were to be simplified as algebraic expressions of order µ − 1.
ABEL never considered whether his definitions of order and degree were total, i.e.
whether any algebraic expression could (uniquely) be ascribed an order and a degree;
throughout his investigations of algebraic expressions, ABEL tacitly used that to any
such object corresponded a unique order and a unique degree. It is obvious that these
concepts introduced a hierarchy on the class of algebraic expressions (see table 6.1).
6.3.2 Standard form
Based on his hierarchy of algebraic expressions, ABEL demonstrated a central theorem
concerning these newly defined objects. It was to serve as a concrete standard form
for algebraic expressions. First, ABEL found a slightly modified standard form (6.1)
by writing an algebraic expression v of order µ and degree m as
�
v = f r1, . . . , rk, p√ �
R , (6.2)
where f was rational, r1, . . . , rk were expressions of order µ but degree at most m − 1,
whereas R was an expression of order µ − 1 such that
p√
R could not be expressed
rationally in r1, . . . , r k, and p was a prime. ABEL obtained this alternative standard
form (6.2) from (6.1) by allowing the arguments r1, . . . , r k to be of the same order as
v, but of lower degree. The two standard forms were equivalent and the hierarchic
structure in the class of algebraic expressions was preserved.
Writing the rational expression f as the ratio of two entire expressions,
v = T
�
r1, . . . , rk, p√ �
R
�
V r1, . . . , rk, p√ �,
R
ABEL specified the form of v as the ratio of two polynomials in p√ R of degree at most
p − 1,
v = T
. (6.3)
V