Proofs of the Fundamental Theorem of Algebra
In his first proof of the Fundamental Theorem of Algebra, Gauss deliberately avoided using imaginaries. When formulated for a polynomial with real coefficients, the theorem states that every such polynomial can be represented as a product of first and second degree terms. Second degree factors correspond to pairs of conjugate complex roots. Over the field of complex numbers a more elegant formulation is possible: every polynomial is a product of first degree terms. The latter formulation is not only more elegant, it's also more revealing. Staying in the realm of real numbers it's hard to explain why and wherefrom quadratic terms appear. Complex numbers provide an immediate explanation.
Complex numbers indeed proved to be a natural setting for the theorem. But the realization of course did not come immediately. The father of the modern complex analysis, A.L.Cauchy (1789-1857), indeed felt comfortabe in the complex domain but the proof we have here utilizes very little the powerful features that come along in transition from real to complex numbers. The proofs by Liouville (1809-1882) and R.P.Boas, Jr. (1912-1992) make a convincing argument that the complex plane and the theory of analytic functions form the natural setting for the theorem.
Real functions may or may not have derivatives. Furthermore, existence of a derivative in one point does not assure its existence anywhere else. A real function may have one or two or, for that matter, any finite number of derivatives. In the complex plane, existence of the limit Δf/ Δz, as Δz approaches 0, leads to a host of features unheard of among real functions. Functions for which this limit (the derivative) exists in every point of an open domain are called analytic (in this domain.) Functions analytic in the whole plane are called entire. Polynomials are entire functions. Analytic functions have derivatives of any order which themselves turn out to be analytic functions. Both real u and imaginary v, components of analytic functions f(z) = u(z) + iv(z) are real valued harmonic functions which, like a travelling wave, are completely defined by their boundary values. For analytic functions, this property is expressed by the Cauchy Integral Formula.
Functions which are not entire have singularities in the finite plane. Entire functions have a singularity at infinity. The only ones that do not are constant. Polynomials have a pole at infinity. All entire functions with a pole of the same order behave in a similar manner. All polynomials of order n behave similarly to zn which has been exploited in somewhat different ways in the proofs by Cauchy and those taken from books by Birkhoff and MacLane and Courant and Robbins. One facet of this property found an expression for more general analytic functions in the form of a theorem proven by the French mathematician Eugene Rouche (1832-1910) in 1883: under certain conditions, the inequality |f(z) + g(z)| < |f(z)| valid for z on a simple closed curve, implies that inside the curve, analytic function f and g have the same number of zeros. Applied to polynomials P(z) of degree n and zn with the curve being a circle of sufficiently large radius, Rouche's theorem yields the Fundamental Theorem of Algebra.