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

112 Chapter 6. Algebraic insolubility of the quintic
2. On the other hand, if a rational function v = v (x1, . . . , xn) satisfies an equation
of degree m with symmetric functions of x1, . . . , xn as its coefficients, the func-
tion v must have at most m different values under permutations of x1, . . . , xn. If
the equation is furthermore known to be irreducible, v must take on exactly m
values. Thus, to the relation v = m√ R corresponds an equation of degree m with
symmetric coefficients.
6.6 ABEL’s combination of results into an impossibility
proof
The fourth component of ABEL’S impossibility proof concerned detailed and highly
explicit, “computational” investigations of functions of five quantities having two or
five values. ABEL sought to reduce all such functions to a few standard forms, an
approach completely in line with the classification which opened his paper. These
investigations have been subjected to quite a lot of criticism, rethinking, and eventu-
ally incorporation into a broader theory, all of which will be dealt with in subsequent
chapters.
6.6.1 Careful studies of functions of five quantities
The CAUCHY-RUFFINI theorem described in sections 5.6 and 6.4 had ruled out the ex-
istence of functions of five quantities which had three or four different values when
their arguments were permuted. The remaining relevant (non-symmetric) cases were
concerned with functions having two or five values. In the case of two-valued func-
tions, ABEL reduced all such functions to the alternating one which CAUCHY had also
studied; and when the function had five values, ABEL could write it as a fourth degree
polynomial in one of the variables with coefficients symmetric in the remaining four.
Two-valued functions. In order to describe functions of five quantities having two
values under permutations, ABEL let v denote such a function of x1, . . . , x5 having
the two values v1, v2. Furthermore, he let v ′ denote a second such function (with the
values v ′ 1 and v′ 2 ) and formed two further functions
t1 = v1 + v2, and
t2 = v1v ′ 1 + v2v ′ 2 .