The proof that Vaughan outlined is found in almost exactly the same form in Courant & Robbins, published around 60 years ago. Here is a proof I learned in grad school, based on three facts, one analytic and two algebraic. The analytic fact, which is an irreducible minimum, given that the reals cannot be defined algebraically (except as a real closed field of continuum transcendence degree, which misses the point) is the intermediate value theorem. In other words, order completeness. From this follows the fact that every odd order polynomial has a (real) root and that every real number--and with a bit of manipulation, every complex number--has a complex square root. The algebraic facts are the existence of a splitting field and the theorem on elementary symmetric functions, neither of which is quite trivial, nor very deep. The way you do it is by proving that every real polynomial of degree n = 2^k*m with m odd has a complex root, by induction on k. The case k = 0 is quite trivial, of course. So suppose that f is a real polynomial of even degree n and, in some splitting field has roots r_1,...,r_n. For each integer s form the polynomial f_s = \prod{i<j}(x - r_i - r_j - sr_ir_j) which has degree n(n-1)/2, which is less 2 divisible than n. The theorem on symmetric functions implies it is real and hence for some i and j dependent on s, r_i + r_j + sr_ir_j is in C. Since there are only finitely many pairs i and j and infinitely many integers, there are distinct s and t for which both r_i + r_j + sr_ir_j and r_i + r_j + tr_ir_j belong to C. Given that C has square roots, one easily discovers that r_i and r_j are both complex numbers. To go from real to complex polynomials, just multiply a complex polynomial by its conjugate; for a root r of the product, one of r and its conjugate is a root of the original. And the process of factoring completely is well-known. Incidentally, it is the case that if K is an algebraic extension of L and every polynomial with coefficients in L has at least one root in K, then K is an algebraic closure of L. The argument above pretty much does the characteristic 0; the prime case is trickier. This is not entirely elementary, but then neither is the winding number argument. Comparing them is difficult because the algebraic argument I have given could all be taught in a lower division course, while the winding number argument, intuitively appealing, really requires some sophisticated stuff to deal with rigorously. Which raises an interesting question. If we agree, as we seem to, that proof is the essence of what we mean by mathematics, then what does it mean to give an intuitively appealing non-proof and call it a proof? Michael