The Fundamental Theorem of Algebra is deeply woven into the fabric of many aspects of
mathematics, from which have emerged proofs with greatly differing viewpoints. Two such
proofs, which are particularly well-known, are described briefly below, with references where
details can be found.
It is proved in the theory of Complex Variables that for a differentiable function
f: C
C,
where the integral is taken over a suitable closed curve enclosing a region where f and
its derivatives are defined and continuous. By means of careful limiting arguments, it follows
that the values of a differentiable function on a suitable region of C are determined
by its values on the boundary of the region, according to the Cauchy Integral Formula:
From this estimate, one obtains Liouville’s Theorem: a bounded function
differentiable on the entire complex plane is constant.
Now consider a non-constant polynomial P(z) of degree n, and suppose it has
no roots. Then the reciprocal 1/P(z) is a continuous, differentiable function on the
entire complex plane. Since, as we have shown above, if P(z) has no roots,
|P(z)| takes on a global minimum value, 1/|P(z)| is bounded. By Liouville’s Theorem
1/P(z), and hence P(z), must be a constant, contradicting our choice of
P(z). This proves the Fundamental Theorem of Algebra. (See, for example, Boas, RP, Invitation to Complex Analysis, Random House, New York, 1987, for details.)
Copyright © 1996-2008 Alexander Bogomolny