On Wed, 29 Mar 2006, Vaughan Pratt wrote:
a few decades ago an elementary exposition of the Fundamental Theorem of Algebra would not be expected to include an elementary proof since the extant proofs were either lengthy arguments or nonelementary appeals to the minimum modulus principle, properties of holomorphic functions such as Liouville's theorem, or other results the reader would be unlikely to be on top of. The dominant belief was that the only short proofs were nonelementary ones.
But for an audience aware only that z^i for any nonnegative integer i maps circles at the origin to i-fold circles of radius r^i at the origin, an entirely elementary notion, an expositor today would be morally obligated to include a full proof since there is hardly anything left to explain. The polynomial a_d z^d + ... + a_0 maps little circles to the neighborhood of a_0 and big circles to a loop tending to a very big d-fold circle of radius a_d r^d, whence the smoothly growing image, under the polynomial, of a smoothly growing circle is obliged to cross the origin at some stage. Still a topological argument, but now an entirely elementary one.
Except, that is, for the theorem that a loop wound d times around the hole in the punctured plane cannot be continuously retracted to a point, which was tacitly smuggled in there. But that statement is less intimidating than anything based on holomorphic functions.
This slick proof seems only to have emerged in the past couple of decades.
Has it? It seems to me no more than (an explicity homotopy-theoretic formulation of) the (implicitly homotopy-theoretic) proof via Rouch\'e's Theorem, which I was taught as an undergraduate (and which I've taught to undergraduates on many occasions since then). Peter Johnstone