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A simple proof of the Fundamental Theorem of Algebra

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HAL Id: hal-03109761

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A simple proof of the Fundamental Theorem of Algebra

Ricardo Pérez-Marco

To cite this version:

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A SIMPLE PROOF OF THE FUNDAMENTAL THEOREM OF ALGEBRA

RICARDO P´EREZ-MARCO

Abstract. We present a simple short proof of the Fundamental Theorem of Algebra, without complex analysis and with a minimal use of topology.

1. Statement.

Theorem 1.1. A non constant polynomial P (z) ∈ C[z] with complex coefficients has a root.

The proof is based only on the following elementary facts: • A polynomial has at most a finite number of roots. • The Implicit Function Theorem.

• Removing from C a finite number of points leaves an open connected space.

2. The proof.

It is enough to consider a monic polynomial P . We denote by C = (P′

)−1(0) the finite set of

critical points of P , and by D = P (C) the finite set of critical values of P .

• Let R = {c ∈ C; the polynomial P (z) − c has at least a simple root and no double roots}. • R ⊂ C − D. This is because if c ∈ D, then c = P (z0) for some critical point z0 ∈ C, hence

P′

(z0) = 0 and P (z) − c = 0 has a double root at z0. Note that C − D is open and connected (D

being finite).

• R is open. This is an application of the Implicit Function Theorem. Let c0 ∈ R ⊂ C − D,

and z0 ∈ C be a root of P (z) − c0. We apply the Implicit Function Theorem to the equation

F (z, c) = P (z) − c = 0. Since ∂F

∂z(z0, c0) = P ′

(z0) 6= 0, there is a neighborhood U of c0such that for

c ∈ U we have a root z(c) of P (z) − c. Taking U small enough, by continuity of P′

and c 7→ z(c), we have P′

(z(c)) 6= 0 and the root z(c) is simple. Since C − D is open we can take U ⊂ C − D and P (z) − c does not have any double root, thus U ⊂ R.

• R is closed in C − D. Because P is monic, if c is uniformly bounded then any root of P (z) − c is uniformly bounded (since P (z)/zn

→ 1 uniformly when z → ∞, if n is the degree). We can take a subsequence of cn→ c∞ ∈ C− D and a converging subsequence of roots of P (z)− cn. By continuity,

the limit is a root of P (z) − c∞, so this polynomial has roots. Moreover, all roots of P (z) − c∞ are

simple since c∞∈ C − D.

2010 Mathematics Subject Classification. 30C15, 12D10.

Key words and phrases. Roots, complex polynomials, fundamental theorem of algebra.

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2 R. P ´EREZ-MARCO

• R is non-empty. For any a ∈ C we have that for c = P (a), P (z) − c has at least z = a as root. If we choose a ∈ C − P−1

(D), then for any root z0 of P (z) − c with c = P (a), we have

P (z0) = P (a) /∈ D, so z0∈ P/ −1(D), but C ⊂ P−1(D), and z0∈ C, and the root z/ 0 is simple.

The above proves that R = C − D. Now, if 0 ∈ D, then 0 = P (z0) for a critical point z0 of P

that is also a root of P . If 0 /∈ D, then 0 ∈ R = C − D and the equation P (z) − 0 = 0 has a simple root. In all cases P has a root. ⋄

3. Comment.

The above proof is inspired from a beautiful proof by Daniel Litt [1]. He works in the global space of monic polynomials of degree n ≥ 1 (biholomorphic to Cn), and removes the algebraic locus

Dn, defined by the discriminant, of polynomials with a double root. He uses that the complement

of an algebraic variety in Cn is connected. Essentially the proof above achieves the same goal in

a more elementary way working with n = 1. In particular, we only need the simpler fact that the complement of a finite set in the plane is connected (which for n = 1 is the same as the connectedness of the complement of an algebraic variety in Cn

). We also avoid the use of discriminants.

Acknowledgment. I am grateful to my friends Marie-Claude Arnaud, Kingshook Biswas, Alain Chenciner and Yann Levagnini for their comments and suggestions to improve the presentation. In particular, to Kingshook that proposed a simplification of a first draft.

References

[1] LITT, D.; Yet another proof of the Fundamental Theorem of Algebra, Manuscript, 2011. (www.daniellitt.com/blog/2016/10/6/a-minimal-proof-of-the-fundamental-theorem-of-algebra) CNRS, IMJ-PRG, Unversit´e de Paris, Boˆıte courrier 7012, 75005 Paris Cedex 13, France Email address: ricardo.perez.marco@gmail.com

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