> Is it possible to prove the fundamental theorem of
> algebra strictly algebraically, with no element of analysis?
>
Because of the very statement, analysis can't be entirely eliminated. I
recall a proof in MacDuffee's algebra textbook (circa ~1945) that comes
pretty close. The irreducible piece of analysis (if I remember correctly
half a century later) is the theorem that a polynomial of odd degree with
real coefficients has a real root.
Martin Davis