Theo de Jong gives in the Monthly of November 2009
a proof of the fundamental theorem of algebra using the Lagrange multiplier method:
For any complex polynomial F, there is a complex number z=x+i y with
F(z) = zn + a1 zn-1 + ... + an = 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:
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 cn converging to 0 for which
Q=cn is nonsingular and contains a point (an,bn) with P(an,bn)=0.
The sequence (an,bn) is bounded because P2 + Q2 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) = Q2(x,y) has a global minimum on the contraint g(x,y) = P(x,y) = c
because P2 + Q2 goes to infinity for (x,y) to infinity. Such a global minimum satisfies
The Cauchy-Riemann equations Qx = - Py , Qy = Px
which hold because F is analytic allow to rewrite this as
the Lagrange equations
(fx,fy) = L (gx,gy)
g = c
2 Q (Qx,Qy) = L ( Px,Py)
g = c
2 Q (-Py,Px) = L ( Px,Py)
g = c
The first relation tells that either Q=0 or ( Px,Py) 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,