440 15. MODERN ALGEBRATo explain the most important of these new insights, let us consider what
Girard's result means when applied to Cardano's solution of the cubic y^3 + py = q.
If the roots of this equation arc r, s, and t, then ñ = st + tr + rs, q = rst, t= -r-s,
since the coefficient of y^2 is zero. The sequence of operations implied by Cardano's
formula is
Ñ <1
U
= 3; v=
r
a = \/ u^3 + v^2 ;
y = \/v + a + \/v - a.Girard's work implies that the quantity a, which is an irrational function of the
coefficients ñ and q, is a rational function of the roots r, ,s, and t:
a = ±—L={r - s)(s - t)(t - r);that is, it does not involve taking the square root of any expression containing a
root.
1.3. Newton, Leibniz, and the Bernoullis. In the 1670s Newton wrote a text-
book of algebra called Arithmetica universalis, which was published in 1707, in
which he stated more clearly and generally than Girard had done the relation be-
tween the coefficients and roots of a polynomial. Moreover, he showed that other
symmetric polynomials of the roots could be expressed as polynomials in the coef-
ficients by giving a set of rules that are still known by his name, although Edward
Waring also proved that such an expression is possible.
Another impetus toward the fundamental theorem of algebra came from calcu-
lus. The well-known method known as partial fractions for integrating a quotient
of two polynomials reduces all such problems to the purely algebraic problem of
factoring the denominator. It is not immediately obvious that the denominator
can be factored into linear and quadratic real factors; that is the content of the
fundamental theorem of algebra. Johann Bernoulli (1667-1748, the first of three
mathematicians named Johann in the Bernoulli family) asserted in a paper in the
Acta eruditorum in 1702 that such a factoring was always possible, and therefore
all rational functions could be integrated. Leibniz did not agree, arguing that the
polynomial x^4 + a^2 , for example, could not be factored into quadratic factors over
the reals. Here we see a great mathematician being misled by following a method.
He recognized that the factorization had to be (÷^2 + a^2 \/—)(x^2 — a^2 \f—I) and that
the first factor should therefore be factored as (x + ay/—y/^t)(x — á\/—ô/^ú) and
the second factor as (x + d\/V-l)(i - á\/\/—ú), but he did not realize that these
factors could be combined to yield x^4 + a^2 = (x^2 - \/2ax + a^2 )(x^2 + s/2ax + a^2 ).
It was pointed out by Niklaus Bernoulli (1687 1759, known as Niklaus I) in the
Acta eruditorum of 1719 (three years after the death of Leibniz) that this last
factorization was a consequence of the identity x^4 + a^4 = (x^2 + a^2 )^2 - 2a^2 x^2.
1.4. Euler, d'Alembert, and Lagrange. The eighteenth century saw consid-
erable progress in the understanding of equations in general and the procedures
needed to solve them. Much of this new understanding came from the two men
who dominated mathematical life in that century, Euler and Lagrange.