Perhaps the moral is not to bother with the Britannica. Wikipedia has several proofs including the winding number argument and the one I outlined using the symmetric function argument. Then a couple of analytic ones. Of course, Wiki has no size limitations. Perhaps we have flogged this particular horse enough. On Sun, 2 Apr 2006, Vaughan Pratt wrote:
Fred E.J. Linton wrote:
First, in Birkhoff & Mac Lane (my own undergraduate algebra text), Section 3 of Chapter V of the 1953 ("revised") edition offers a proof along winding number lines on pp. 107-109.
Thanks, Fred, I wish I'd noticed that before. I have the sixth printing (1948) of the 1941 edition, which says, "Many proofs...are known; ...we have selected one whose non-algebraic part is *especially plausible intuitively*." (My emphasis.) Then they give the proof "I like".
To administer one more lash to this dead horse, the wording in the Britannica article implies that the absence of an elementary algebraic argument was the reason for omission of a proof of FTAlg. Whence the change of heart about arguments that are "especially plausible intuitively?" If they're good enough for an algebra text they should be even more acceptable for an encyclopaedia article.
Vaughan Pratt