The History of Mathematics: A Brief Course

450 15. MODERN ALGEBRA

As elliptic integrals—integrals containing the square root of a cubic or quartic

polynomial—became better understood, both computational and theoretical con-

siderations brought about a focus on transformations of one elliptic integral to

another. In 1828 Jacobi studied rational changes of variable y = U(x)/V(x), where

U and V are polynomials of degree at most n, and found an algebraic equation that

U and V must satisfy in order for this transformation to convert an elliptic integral

containing one parameter (modulus) into another.

The transformation

where u = ^/c and í = í'ë (see Klein, 1884, Part H) Chapter 1, Section 3). Galois

had recognized this connection and noted that the general modular equation of

degree 6 could be reduced to an equation of degree 5 of which it was a resolvent. The

parameter u can be expressed as a quotient of two infinite series (theta functions)

in the number q = e~*K'/K, where Ê and K' are the complete elliptic integrals

of first kind with moduli ê and \/l - ê^2. Thus, a family of equations of degree 6

containing a parameter could be solved using the elliptic modular function. It was

finally Charles Hermite (1822-1901) who, in 1858, made all these facts fit together

in a solution of the general quintic equation using elliptic functions.

Solution of particular quintics by radicals. The study of particular quintics that
are solvable by radicals has occupied considerably more time. It is not difficult to
reduce the general problem to the study of equations of the form x^5 + px + g = 0
via a Tschirnhaus transformation. This topic was studied by Carl Runge (1856-
1927) in an 1886 paper in the Acta mathematica. There are only five groups of
permutations of five letters that leave no letter fixed and hence could be the group
of an irreducible quintic equation. They contain respectively 5, 10, 20, 60, and 120
permutations. A quintic equation having one of the first three as its group will be
solvable by radicals, whereas an equation having either of the other two groups will
not be. The actual construction of the solution, however, is by no means trivial.
The situation is similar to that involved in the construction of regular polygons with
ruler and compass. Thanks to Galois theory, we now know that it is possible for
a person with sufficient patience to construct a 17-sided regular polygon—that is,
partition a circle into 17 equal arcs—using ruler and compass, and Gauss actually
did so.^11 The details of the construction, however, are quite complicated. The
same theory assures us that it is similarly possible to divide the circle into 65,537
congruent arcs, a task attempted by Johann Hermes (see p. 189). In contrast,
algorithms have been produced for solving quintics by radicals where it is possible
to do so.^12 An early summary of results in this direction was the famous book
by Felix Klein on the icosahedron (1884). An up-to-date study of the theory of
solvability of equations of all degrees, with historical documentation, is the book of
R. Bruce King (1996).

(^11) Abel, using elliptic functions, partitioned the lemniscate into 17 arcs of equal length.
(^12) See the paper by D. S. Dummit "Solving solvable quintics," in Mathematics of Computation,
57 (1991), No. 195, 387-401.
corresponds to an equation
u^6 - v^6 + 5u^2 v^2 (u^2 - v^2 ) + 4uu(l - uV) = 0,