Section 56: The Fundamental Theorem of Algebra
The proof is, in fact, rather hard; the most difficult part is to prove that every polynomial equation of positive degree has at least one root... One can use only techniques of algebra; this proof is long and arduous. Or one can develop the theory of analytic functions of a complex variable to the point where it becomes a trivial corollary of Liouville’s theorem. Or one can prove it as a relatively easy corollary of our computation of the fundamental group of the circle...
(The Fundamental Theorem of Algebra) A polynomial of degree
has at least one complex root.
If
is given by
, then
is injective (if
is a single loop, then
), and, hence
is not nulhomotopic (
is injective because
is a retract of
, hence
is injective and non-trivial).
If
, then
so that without loss of generality
.
But then, if there is no root, then
is nulhomotopic (because extendable to
) and homotopic to
(via
which is never zero because
). Contradiction.
- If is such that , then all roots are in the interior of .