As a warm-up, I want to give probably the most beautiful proof of the Fundamental Theorem of Algebra which I know, using the theory of one complex variable. In case you don't know the theorem:
Let be a polynomial,
. If
, there exist constants
,
such that
The main ingredient of the proof is the following statement, which is in fact eqiuvalent to the Fundamental Theorem:
Assume that on the contrary, is zero-free. In that case,
defines a entire function, i.e. a function defined on
which is holomorphic everywhere. We will show that
is bounded, whence it follows by Liouville's Theorem that
is constant. This implies that
is constant, a contradiction.
First, write with
; then
Clearly, for ,
uniformly, whence
uniformly for
. Therefore,
is bounded on
for some
.
Now consider on
. We have that
is continuous on
, and since
is compact, we know that
attains its maximum on
, whence
is bounded on
.
Therefore, is bounded on
, and we can conclude.
Now we are able to prove the Fundamental Theorem:
We proceed by induction on , the degree of
. If
, then
, whence we can set
and
.
Now assume that the statement holds for polynomials of degree . Let
be a polynomial of degree
. By the lemma, there exists some
with
. Now, using the Division Algorithm, write
with polynomials
satisfying
, i.e.
. Now
whence we have and, therefore,
. As
, we have
.
Therefore, by the induction hypothesis, there exist ,
with
, whence
i.e. the induction hypothesis holds for , too.
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