Fundamental Theorem of Algebra
Yet Another Proof
Theorem. The Fundamental Theorem of Algebra. Let
|P(z) = a0zn + a1zn-1 + ... + akzn - k + ... + an|
be a polynomial of degree n ≥ 1 with complex numbers ai as coefficients. Then P has a root, i.e., there is a φ C such that P(φ) = 0.
We prove the theorem by showing that Image(P) = C.
We assume the standard result that a complex polynomial P: C → C is a proper map, i.e., P-1(A) is compact whenever A ⊂ C is compact. (P is continuous, and
Let f: U → R2 be a differentiable map of an open set U ⊂ R2 to R2. A point
With this notation in mind, we first prove
Lemma 1. Let K be the set of critical values of P. Then K and P-1(K) are both finite subsets of C.
Proof: The critical points of P are the points at which P’(z) = 0. Since P’ is a polynomial of degree
Lemma 2. Let X = C \ P-1(K) and
Proof. Lemma 1 ensures that both X and Y are open connected subsets of C. Also, observe that all points in X are regular points of P, i.e., DP(x) is invertible for all
Since P: C → C is proper and C is locally compact, it follows that Image(P) is closed in C.
Let y P(X). Then,
Now, by definition, K ⊂ Image(P) and Lemma 2 tells us that
The crucial idea in this proof is that the plane remains connected after removing finitely many points. All the other results used hold for polynomials from R to R as well.
Our approach to the fundamental theorem of algebra is similar to arguments used in  to investigate proper, smooth maps with non-negative Jacobian between connected, orientable manifolds.
- A. Nijenhuis and R. W. Richardson, Jr., A theorem on maps with non-negative jacobians, Michigan Math. J. 9 (1962) 173—176.