Re Gauss's first proof, Smale pointed out the following (discussed by Weber in the collected papers of Gauss): In order to show that the zero sets of the real and imaginary parts of a complex polynomial intersect, Gauss states (paraphrasing): "If a [polynomial] curve C in the plane enters a region, it must leave it. No one to whom I have explained the meaning of this result doubts it. I will give a proof in a later paper." This seems to mean, for example, that no point has an open neighborhood in the plane meeting C in a half open interval, or in a set homeomorphic to 9. In modern terms: (A) For every p in C the number of branches containing p is even. This is true-- but to use it Gauss would have to prove it without using FTA! (A) follows from a vast generalization due independently to D. Sullivan and Deligne: (A') Let p be a point in an analytic variety X over C or R, and S the boundary of a sufficiently small ball centered at p. Then the Euler characteristic of the intersection of S and X is 0 in the complex case, and even in the real case. So Gauss's celebrated proof has an enormous gap.