THE FUNDAMENTAL THEOREM OF ALGEBRA – PROOFS VIA “THE CREEPING LEMMA”

James E. Joseph and Myung H. Kwack

Let ℂ be the complex plane. A complex polynomial function is a function P: ℂ → ℂ defined by

P (z) = ∑_{k=0}^{n} a_{n−k}z^{n−k},

where a_{0}, ⋯, a_{n} ∈ ℂ. The Fundamental Theorem of Algebra states that a nonconstant complex polynomial function has a zero. Over the years, a number of proofs of this theorem have been discovered [1]. In this note, the so-called “Creeping Lemma” [3] is used to present two elementary proofs of the Fundamental Theorem. The proofs can be presented in a first analysis course (or advanced calculus course) as a nice application of diffeomorphism and the concept of least upper bound.

It is known that a complex polynomial function P is a closed function, i.e. that P (A) is a closed subset of ℂ when A is a closed subset of ℂ. The proof for non-constant P combines the Bolzano-Weierstrass property of ℂ with the continuity of P and the observation that |P (z)| gets arbitrarily large as |z| gets arbitrarily large [2].

Here the “Creeping Lemma” is applied

(i) to provide a direct proof of The Fundamental Theorem of Algebra, and (ii) to prove that a non-constant complex polynomial function is an open

function, so P (ℂ) = ℂ, since P (ℂ) is a nonempty, closed, and open subset of the connected space ℂ (another proof of the Fundamental Theorem of Algebra).

In the sequel, the set of critical points of a complex polynomial P, {z ∈ ℂ : P′(z) = 0}, will be denoted by M.

\underline{Theorem 1}. The Fundamental Theorem of Algebra: A non-constant complex polynomial function has a zero.

\underline{Proof}. The set of critical points, M and its image P (M) are both finite. Let p = P (a) with a ∉ M. Let σ: [0, 1] → ℂ be a curve such that σ(0) = p, σ(1) = O, and σ([0, 1)) ⊂ ℂ − P (M), an open connected set. It will be enough to show that there is a curve σ̃: [0, 1] → ℂ with σ̃(0) = a and P ∘ σ̃ = σ.

Choose a neighborhood U of a such that P: U → P (U) is a diffeomorphism with inverse g. Then, σ̃ = g ∘ σ is defined on [0, t] for some t ∈ (0, 1]. Let c = sup{r ∈ (0, 1] : σ̃ can be extended to a curve from [0, r] into ℂ such that σ̃(0) = a, P ∘ σ̃ = σ}. Since σ(c) = lim P ∘ σ̃(s_{k}) for a sequence s_{k} → c and P is a closed function, it may be assumed, without loss of generality, that σ̃(s_{k}) → b ∈ ℂ and P (b) = σ(c). Suppose c < 1. Then, P′(b) ≠ 0 and the point b has a neighborhood V such that P: V → P (V) is a diffeomorphism with inverse g_{b}. The curve g_{b} ∘ σ is defined on (c − ϵ, c + ϵ) ⊂ (0, 1) for some ϵ > 0 and ultimately g_{b} ∘ σ(s_{k}) = σ̃(s_{k}). Note that any curve α: (c − ϵ, c) → ℂ with α((c − ϵ, c)) ∩ V ≠ ∅ is uniquely determined from the condition P ∘ α = σ by α = g_{b} ∘ σ. It follows that σ̃ can be extended to a curve from [0, c + ϵ/2], which is a contradiction. Hence, c = 1 and P (b) = σ(1) = O. The proof is complete.

\underline{Remark}. At a non-critical point z of a polynomial P, the Inverse Function Theorem guarantees that there is an arbitrarily small open neighborhood of z whose image is a neighborhood of P (z). A slight modification of the above proof establishes that at a critical point z of P, there is also an arbitrarily small open neighborhood of z whose image is a neighborhood of P (z). Theorem 2 follows from these observations, and establishes that, for non-constant P, P (V) is an open subset of ℂ for each open V ⊂ ℂ, not only for V = ℂ.

\underline{Theorem 2}. A nonconstant complex polynomial function P is an open function.

\underline{Proof}. Let V ⊂ ℂ be open and let a ∈ V. If P′(a) ≠ 0, P is a diffeomorphism on an open neighborhood U ⊂ V of a and P (V) contains an open neighborhood P (U) of P (a). Suppose P′(a) = 0. Choose a disk D_{s}(a) ⊂ V with P^{−1}(P (a)) ∩ \overline{D}_{s}(a) = \{a\} and D_{s}^{∗}(a) ⊂ ℂ − M, where D_{s}(p) = {q ∈ ℂ : |p − q| < s}, D_{s}^{∗}(p) = D_{s}(p) − {p}. Choose a number r > 0 satisfying the properties

(i) 2r is smaller than the distance from P (a) to P (∂D_{s}(a)), where ∂D_{s}(a) is the boundary of D_{s}(a), and

(ii) D_{r}^{∗}(P (a)) ⊂ ℂ − P (M).

Since P is not constant, choose a point b ∈ D_{s}(a) such that P (b) ∈ D_{r}^{∗}(P (a)). Let q ∈ D_{r}^{∗}(P (a)) and let σ: [0, 1] → D_{r}^{∗}(P (a)) be a curve with σ(0) = P (b) and σ(1) = q. As in the proof of Theorem 1, there is a curve σ̃: [0, 1] → ℂ determined by the conditions σ̃(0) = b and P ∘σ̃ = σ. However, σ̃([0, 1]) ⊂ D_{s}(a), since b ∈ σ̃([0, 1]) ∩ D_{s}(a), and σ([0, 1]) ∩ P (∂D_{s}(a)) = ∅ implies σ̃([0, 1]) ∩ ∂D_{s}(a) = ∅. It follows that q = P ∘ σ̃(1) ∈ P (D_{s}(a)) and thus, D_{r}(P (a)) ⊂ P (D_{s}(a)). The proof is complete.

\underline{References}

1. A. Sen, “Fundamental Theorem of Algebra – Yet Another Proof,” American Mathematical Monthly, 107 (2000), 842–843.

2. J. E. Joseph and M. H. Kwack, “A Note On Closed Functions,” Missouri Journal of Mathematical Sciences, 18 (2006), 61–63.

3. R. M. F. Moss and G. T. Roberts, “A Creeping Lemma,” American Mathematical Monthly, 75 (1968), 649–652.

Mathematics Subject Classification (2000): 30C10, 30C15, 57R50

James E. Joseph Department of Mathematics Howard University Washington, D. C. 20059 email: jjoseph@howard.edu

Myung H. Kwack Department of Mathematics Howard University Washington, D. C. 20059 email: mkwack@howard.edu