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 - r_{1} )(z - r_{2} )...(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
P(z) = a_{n}z^{ n} + a_{n -1}z^{ n -1} + ... + a_{1}z^{1} + a_{0}
behaves essentially like a_{n}z^{ n} whenever |z| is very large, and behaves essentially like a_{1}z^{1} + a_{0} (or a_{m}z^{ m} + a_{0}, 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.
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