Remarks on Proving The Fundamental Theorem of Algebra

Scott E. Brodie

September 28, 1997

The "Fundamental Theorem of Algebra" (FTA) states that every polynomial P(z) of degree n with coefficients drawn from the set C of complex numbers can be factored into the product of n linear factors (z - r₁ )(z - r₂ )...(z - r_n ), where the roots r_k are complex numbers satisfying P(r_k ) = 0.

It is instructive to contemplate various alternatives to the hypotheses of the FTA. The theorem certainly does not hold for the set of real numbers R – for example, the equation x ² + 1 = 0 has no real roots. Likewise, it is insufficient to consider only rational numbers for the coefficients and roots: the equation x ² - 2 = 0 has no rational roots. The theorem applies to polynomials – a transcendental function may have no roots, such as the familiar exponential function e ^z.

These counterexamples suggest that a proof of the FTA must incorporate references to both algebraic properties of polynomials and analytic properties of R or C. Many proofs of the theorem are known, which place varying degrees of emphasis on the algebraic and analytic aspects of the theorem. Below, we sketch two fairly intuitive proofs; two well-known, but technically more demanding lines of argument are mentioned subsequently.

Actually, we argue in each case that the polynomial has at least one root. The conclusion that the number of roots equals the degree of the polynomial (counting multiple roots according to their multiplicity) then follows by a nice argument by mathematical induction. The theorem is obviously true for polynomials of degree one. If the theorem holds for polynomials of degree less than n, consider a polynomial of degree n. It has at least one root, say, r. Thus, we may factor the polynomial into the form P(z) = (z - r)P_n -1 (z), where P_n -1 (z) is a polynomial of degree n -1, which has exactly n -1 roots by the induction hypothesis. Thus the original polynomial has n roots as claimed.

The proofs by Birkhoff and MacLane and by Cauchy both depend on the observation that the polynomial

behaves essentially like a_nz ⁿ whenever |z| is very large, and behaves essentially like a₁z¹ + a₀ (or a_mz ^m + a₀, where a_m is the coefficient with the lowest index m which is not equal to zero) whenever |z| is very small. (This is one point where we observe a critical distinction between polynomials and transcendental functions, which, in essence, contain terms of arbitrarily high degree in their power-series expansions.)

For the sake of completeness, full details of the second proof have been provided.

• Perfect numbers are complex, complex numbers might be perfect
• Fundamental Theorem of Algebra: Statement and Significance
• What's in a proof?
• More about proofs
• Axiomatics
• Intuition and Rigor
• How to Prove Bolzano's Theorem
• Early attempts
• Proofs of the Fundamental Theorem of Algebra
  • Remarks on Proving The Fundamental Theorem of Algebra
    • Sketch of a Proof by Birkhoff and MacLane
    • Sketch of Second Proof (after Cauchy)
    • Details of the Cauchy's Proof
    • Note on the Extreme Value Theorem
    • Sketch of Proof by the methods of the theory of Complex Variables (after Liouville)
    • Maximum Modulus Theorem
  • A Proof of the Fundamental Theorem of Algebra: Standing on the shoulders of giants
  • Yet Another Proof of the Fundamental Theorem of Algebra
  • Fundamental Theorem of Algebra - Yet Another Proof
  • A topological proof, going in circles and counting