Even in my original posting starting this thread I acknowledged that contractibility of the circle was not elementary:
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.
I haven't at any time claimed that it was not necessary to prove this, nor that the proof was easy. What I have been claiming is that the result has a certain self-evident quality to it that, it seemed to me, qualified the argument as at least sufficiently "morally elementary" as to qualify it for inclusion in the Britannic article on algebra. How could the definitive encyclopedia article on algebra not give at least a hint as to why that subject's fundamental theorem was true? However I've been reflecting on just what is behind the very uniform insistence on the distinction between an algebraic proof and an analytic one. Since algebra is descended from analysis, it seems unkind for algebra to deny its parentage in this way. But I see now that this denial is logically necessary. For consider the algebraic plane, the least algebraically closed subfield of the complex plane, consisting of the algebraic numbers. The FTAlg is by definition true there, so it ought to be provable there. One can carry out the same proof, and it all goes through in the same way (using circles of growing algebraic radius, all of which are dense in their complex completion to a connected circle) right up to the last step when we claim that the wildly growing loop that is the image of the tamely growing circle must eventually collide with the origin, d times in fact for a degree d polynomial. And indeed it does, all d times, exactly as with the complex numbers, and with the same roots (the coefficients of the polynomial necessarily being algebraic in this domain). But now analysis has nothing to do with it, since these circles and their image loops while dense are totally disconnected. For all we know the origin could have missed the loop by going through any of its uncountably many gaps. Indeed the loop has measure zero, so the chances of the origin colliding with it even once are less than Buckley's. But with aim that would be the envy of any sniper the origin hit the loop with every one of its d shots. And how do we know this? Using analysis. The consensus would seem to be that there is no other way. Logic alone cannot help. If that's the case, then without analysis there is no algebraic plane. Without the huge continuum to support it, that tiny countable set would not exist! It is ironic that a theorem of algebra about an algebraic domain that itself has no element of analysis to it, being just the algebraic closure of the rationals, a small and totally disconnected space, should require analysis, the parent of algebra, for its proof. The fundamental theorem of algebra is like a student calling home for more money. It takes a continuum to raise an algebraic number. Vaughan Pratt