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Is there a purely algebraic proof of the Fundamental Theorem of Algebra?
I have heard many times that there is no algebraic proof of the fundamental theorem of algebra (and so people say that it is not a theorem of algebra and also for modern algebra it is not so fundamental. However I was wondering if there was a proof that there is no algebraic proof.
I have had a look but I can't seem to find anything online (I may just be missing something). When my lecturer was asked he said that he was not sure but that intuitively if there was one then it should hold for other fields (such as $\mathbb{R}$) but clearly it doesn't. I have also read other places that you need the completness of the reals or something like it. So I was wondering is there a proof that it cannot be proved using purely algebra?
Thanks for any help
