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

6.3. Classification of algebraic expressions 101
2. The classification also enabled ABEL to link the number of values under permu-
tations to the exponent of the involved root extraction.
3. By adapting CAUCHY’S theory of permutations, a restriction of the possible num-
ber of values under permutations to 2 or 5 was achieved.
4. Finally, ABEL reduced each of the possible cases by indirect proofs.
In general, ABEL used references in accordance with the nineteenth century tradi-
tion. Throughout, ABEL’S approach to the question of solubility of the quintic was
based on counting the number of values which a rational function took when its ar-
guments were permuted. Thus, he clearly worked in the tradition initiated by J. L.
LAGRANGE (1736–1813), and it is a little remarkable that no reference to — or even
mention of — LAGRANGE was ever made in ABEL’S published works on the theory
of equations. I take this as an indication that during the half-century elapsed since
LAGRANGE’S trend-setting research, 14 his results and approach had become common
practice in the field. On the other hand, ABEL made explicit reference to CAUCHY’S
work on the theory of permutations, 15 from which he had borrowed the CAUCHY-
RUFFINI theorem without proof in his original 1824 version. 16 In the proof published
two years later in CRELLE’S Journal für die reine und angewandte Mathematik, 17 ABEL
provided the theorem with his own shorter proof, keeping the reference. Thus, by the
same argument as above, CAUCHY’S much younger theory had not yet been as widely
established.
6.3 Classification of algebraic expressions
The objects which ABEL called algebraic functions — and which I term algebraic expres-
sions — were explicit algebraic functions: finite combinations of constant and variable
quantities obtained by basic arithmetical operations. If the operations included only
addition and multiplication, the expression was said to be entire; if, furthermore, di-
vision was involved, it was called rational; and if, additionally, root extractions were
allowed, the expression was denoted an algebraic expression. Subtraction and extraction
of roots of composite degree were explicitly reduced to addition and the extraction of
roots of prime degree, respectively, in order to be contained in the above operations.
In the subsequent classification, ABEL benefited from the simplicity introduced by this
minimal definition in which only root extractions of prime degree were considered.
The purpose of ABEL’S investigations of algebraic expressions was to obtain an im-
portant auxiliary theorem for his impossibility proof. Based on a definition which in-
14 (Lagrange, 1770–1771).
15 (A.-L. Cauchy, 1815a).
16 (N. H. Abel, 1824b).
17 (N. H. Abel, 1826a).