438 15. MODERN ALGEBRAfactions as there are numbers given." He noted that the number of terms in each
faction could be found by using Pascal's triangle.
Girard always regarded the leading coefficient as 1. Putting the equation into
this form, he stated as a theorem (see, for example, Struik, 1986, p. 85) that "all
equations of algebra receive as many solutions as the denomination [degree] of the
highest form shows, except the incomplete, and the first faction of the solutions is
equal to the number of the first mixed [that is, the cofficient of the power one less
than the degree of the equation], their second faction is equal to the number of the
second mixed, their third to the third mixed, and so on, so that the last faction is
equal to the closure [product], and this according to the signs that can be observed
in the alternate order." This recognition that the coefficients of a polynomial are
elementary symmetric polynomials in its zeros was the first ray of light at the dawn
of modern algebra.
By "incomplete," Girard meant equations with some terms missing. In some
cases, he said, these may not have a full set of solutions. He gave the example of the
equation x^4 = 4x — 3, whose solutions he gave as 1, 1, —1 + \/-2, and -1 — \f-2,
showing that he realized the need to count both complex roots and multiple real
roots for the sake of the general rule. He invoked the simplicity of the general rule
as justification for introducing the multiple and complex roots, along with the fact
that complex numbers provide solutions where otherwise none would exist.1.2. Tschirnhaus transformations. Every complex number has nth roots—
exactly ç of them except in the case of 0—that are also complex numbers. As
a consequence, any formula for solving equations that involves only the applica-
tion of rational operations and root extractions starting with the coefficients will
remain within the domain of complex numbers. This elementary fact led to the
proposition stated by Girard, which we know as the fundamental theorem of alge-
bra. Finding such a formula for equations of degree five and higher was to become
a preoccupation of algebraists for the next two centuries.
Analysis of the cubic. By the year 1600 equations of degrees 2, 3, and 4 could all be
solved, assuming that one could extract the cube root of a complex number. The
methods used suggest an inductive process in which the solution of an equation of
degree n, say
xn - áé÷71-1 + · · · Ô an~]X ± a„ = 0,would be found by a substitution y — x"~* - b\Xn~^2 + • · · ± (^6) n-2X Ô l>n-\ with
the coefficients bi,..., 6„_i chosen so that the original equation becomes yn = C.
Observe that there are ç — 1 coefficients bjt at our disposal and ç — 1 coefficients
oi,..., áç_é to be removed from the original equation. The program looks feasible.
Something of the kind must have been the reasoning that led Ehrenfried Walther
von Tschirnhaus (1652 1708) to the belief that he had discovered a general solution
to all polynomial equations. In 1677 he wrote to Leibniz:
In Paris I received some letters from Mr. Oldenburg, but from lack
of time have not yet been able to write back that I have found a
new way of determining the irrational roots of all equations... The
entire problem reduces to the following: We must be able to remove
all the middle terms from any equation. When that is done, and as
a result only a single power and a single known quantity remain,
one need only extract the root.