
Chapter 2
The Fundamental Theorem of Algebra
At the time of Galois, the natural setting for most mathematical investigations was the
complex number system. The real numbers were inadequate for many questions, be-
cause −1 has no real square root. The arithmetic, algebra, and—decisively—analysis
of complex numbers were richer, more elegant, and more complete than the corre-
sponding theories for real numbers.
In this chapter we establish one of the key properties of C, known as the Fun-
damental Theorem of Algebra. This theorem asserts that every polynomial equation
with coefficients in C has a solution in C. This theorem is, of course, false over R—