Question:
Is there already in the literature a proof of the fundamental theorem of algebra as a consequence of Brouwer's fixed point theorem?
N.B. The original post contained superfluous information, but it did generate one answer with a source that claims such a proof is impossible, and another answer with a source that claims to carry out precisely such a proof. Clearly these cannot both be correct.
Best Answer
With regard to the answer already provided:
The Arnold proof is well known to be erroneous, but a correct (as far as I know) version is cited in an earlier MO post here. In particular, it is a proof of the FTA via the Brouwer Fixed Point Theorem.
The latter source is:
Edit 1: Todd Trimble has kindly provided a link to the Fort paper that does not require JSTOR access.
Separately, I see the following quotation:
"Recently, there have been very interesting proofs of the Brouwer theorem. Kulpa deduced a generalization of the Brouwer theorem from the Fubini theorem and the Weierstrass approximation theorem, and applied it to give a simple proof of the fundamental theorem of algebra."
The source of this excerpt is:
And the reference under discussion is:
I gathered this information at the request of D. Goroff some time ago, at which point my search for the Kulpa paper was, unfortunately, fruitless.
Edit 2: Karol Szumiło remarks that a friend in one of Warsaw's libraries was able to track down the Kulpa paper! Given the difficulty of finding it, I have uploaded a copy here.