Theo de Jong gives in the Monthly of November 2009
a proof of the fundamental theorem of algebra using the Lagrange multiplier method:
Theorem:
For any complex polynomial F, there is a complex number z=x+i y with
F(z) = z^{n} + a_{1} z^{n1} + ... + a_{n} = 0 .

In other words, when writing F(x,y) = P(x,y) + i Q(x,y), there is a point (x,y), where the real functions
P and Q are both simultaneously zero. The theorem follows from the following lemma:
Lemma:
If the level curve P= c contains no critical points of P, then it contains a point (a,b) with
Q(a,b) = 0.

Proof of the fundamental theorem:
There are infinitely many values of c, where the level curve P=c is not empty with no critical points of P.
Looking at such a level curve P=c, the lemma implies that the level curve Q=0 is not empty.
If Q=0 contains no critical points of Q, the lemma (applied with reversed P and Q)
implies that it contains a point (a,b) with P(a,b)=0 proving the theorem.
If Q=0 contains a critical point, there is a sequence of points c_{n} converging to 0 for which
Q=c_{n} is nonsingular and contains a point (a_{n},b_{n}) with P(a_{n},b_{n})=0.
The sequence (a_{n},b_{n}) is bounded because P^{2} + Q^{2} goes to infinity at infinity.
There is therefore an accumulation point (a,b). This is the simultaneous root of P and Q.
Proof of the lemma: The function f(x,y) = Q^{2}(x,y) has a global minimum on the contraint g(x,y) = P(x,y) = c
because P^{2} + Q^{2} goes to infinity for (x,y) to infinity. Such a global minimum satisfies
the Lagrange equations
(f_{x},f_{y}) = L (g_{x},g_{y})
g = c

which is
2 Q (Q_{x},Q_{y}) = L ( P_{x},P_{y})
g = c

The CauchyRiemann equations Q_{x} =  P_{y} , Q_{y} = P_{x}
which hold because F is analytic allow to rewrite this as
2 Q (P_{y},P_{x}) = L ( P_{x},P_{y})
g = c
The first relation tells that either Q=0 or ( P_{x},P_{y}) is parallel
to a vector perpendicular to it, a case not possible because P=c was assumed
to have no critical point. There is therefore a point (a,b) on the level curve,
where Q(a,b)=0.
