Chapter 15. Modern Algebra
By the mid-seventeenth century, the relation between the coefficients and roots of a
general equation was understood, and it was conjectured that if you counted roots
according to multiplicity and allowed complex roots, an equation of degree ç would
have ç roots. Algebra had been consolidated to the point that the main unsolved
problem, the solution of equations of degree higher than 4, could be stated simply
and analyzed.
The solution of this problem took nearly two centuries, and it was not until the
late eighteenth and early nineteenth centuries that enough insight was gained into
the process of determining the roots of an equation from its coefficients to prove
that arithmetic operations and root extractions were not sufficient for this purpose.
Although the solution was a negative result, it led to the important concepts of
modern algebra that we know as groups, rings, and fields; and these, especially
groups, turned out to be applicable in many areas not directly connected with
algebra. Also on the positive side, nonalgebraic methods of solving higher-degree
equations were also sought and found, and a theoretically perfect way of deciding
whether a given equation can be solved in radicals was produced.
1. Theory of equations
Viete understood something of the relation between the roots and the coefficients
of some equations. His understanding was not complete, because he was not able
to find all the roots. Before the connection could be made completely, there had to
be a domain in which an equation of degree ç would have ç roots. Then the general
connection could be made for quadratic, cubic, and quartic equations and general-
ized from there. The missing theorem was eventually to be called the fundamental
theorem of algebra.^11.1. Albert Girard. This fundamental theorem was first stated by Albert Girard
(1595-1632), the editor of the works of Simon Stevin. In 1629 he wrote L'invention
nouvelle en I'algebre (New Discovery (Invention) in Algebra). This work contained
some of the unifying concepts that make modern algebra the compact, efficient
system that it now is. One of these, for example, is regarding the constant term
as the coefficient of the zeroth power of the unknown. He introduced the notion
of factions of a finite set of numbers. The first faction is the sum of the numbers,
the second the sum of all products of two distinct numbers from the set, and so
on. The last faction is the product of all the numbers, so that "there are as many(^1) In his textbook on analytic function theory (Analytic Function Theory, Ginn & Co., Boston,
1960, Vol. 1, p. 24), Einar Hille (1894-1980) wrote that "modern algebraists are inclined to deny
both its algebraic and its fundamental character." Hille does not name the modern algebraists,
but he was a careful writer who must have had someone in mind. In the context of its time, the
theorem was both algebraic and fundamental.
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