10 1 History of Classical Algebra
(ii) Viète required “homogeneity” in algebraic expressions: all terms had to be of the
samedegree.Thatiswhythe above quadratic iswritten in what tousis an unusual
way, all terms being of the third degree. The requirement of homogeneity goes
back to Greek antiquity, where geometry reigned supreme. To the Greek way of
thinking, the product ab (say) denoted the area of a rectangle with sides a and b;
similarly, abc denoted the volume of a cube. An expression such as ab + c had
no meaning since one could not add length to area. These ideas were an integral
part of mathematical practice for close to two millennia.
(iii) Another aspect of the Greek legacy was the geometric justification of algebraic
results, as was the case in the works of al-Khwarizmi and Cardano. Viète was no
exception in this respect.
(iv) Viète restricted the roots of equations to positive real numbers. This is under-
standable given his geometric bent, for there was at that time no geometric
representation for negative or complex numbers.
Most of these shortcomings were overcome by Descartes in his important book
Geometry (1637), in which he expounded the basic elements of analytic geome-
try. Descartes’ notation was fully symbolic—essentially modern notation (it would
be more appropriate to say that modern notation is like Descartes’). For example,
he used x,y,z,... for variables and a, b,c, for parameters. Most importantly, he
introduced an “algebra of segments.” That is, for any two line segments with lengths
a and b, he constructed line segments with lengths a +b, a −b, a ×b, and a/b. Thus
homogeneity of algebraic expressions was no longer needed. For example, ab + c
was now a legitimate expression, namely a line segment. This idea represented a
most important achievement: it obviated the need for geometry in algebra. For two
millennia, geometry had to a large extent been the language of mathematics; now
algebra began to play this role. See [1], [7], [10], [12], [17].
1.7 The theory of equations and the FundamentalTheorem
of Algebra
Viète’s and Descartes’ work, in the late sixteenth and early seventeenth centuries,
respectively, shifted the focus of attention from the solvability of numerical equations
to theoretical studies of equations with literal coefficients. A theory of polynomial
equations began to emerge. Among its main concerns were the determination of the
existence, nature, and number of roots of such equations. Specifically:
(i) Does every polynomial equation have a root, and, if so, what kind of root is it?
This was the most important and difficult of all questions on the subject. It turned
out that the first part of the question was much easier to answer than the second.
The Fundamental Theorem of Algebra (FTA) answered both: every polynomial
equation, with real or complex coefficients, has a complex root.
(ii) How many roots does a polynomial equation have? In his Geometry, Descartes
provedtheFactorTheorem,namelythatifα is a root of thepolynomialp(x),then
x−α isafactor,thatis,p(x) = (x−α)q(x), whereq(x)isapolynomialofdegree