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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:

Some Properties of Continuous Functions. M. K. Fort, Jr. The American Mathematical Monthly, Vol. 59, No. 6 (Jun. - Jul., 1952), pp. 372-375. http://www.jstor.org/stable/2306806.

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:

Park, S. (1999). Ninety years of the Brouwer fixed point theorem. Vietnam Journal of Mathematics, 27(3), 187-222. http://www.math.ac.vn/publications/vjm/vjm_27/No.3/187-222_Park.PDF.

And the reference under discussion is:

W. Kulpa, An integral criterion for coincidence property, Radovi Mat.6 (1990) 313-321.

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.

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:

Some Properties of Continuous Functions. M. K. Fort, Jr. The American Mathematical Monthly, Vol. 59, No. 6 (Jun. - Jul., 1952), pp. 372-375. http://www.jstor.org/stable/2306806.

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:

Park, S. (1999). Ninety years of the Brouwer fixed point theorem. Vietnam Journal of Mathematics, 27(3), 187-222. http://www.math.ac.vn/publications/vjm/vjm_27/No.3/187-222_Park.PDF.

And the reference under discussion is:

W. Kulpa, An integral criterion for coincidence property, Radovi Mat.6 (1990) 313-321.

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.

I believe the answer to this question is actually yes, and would be quite pleased if someone could communicate a copy of the last reference I have listed here. (My about-page has an email address.)

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:

Some Properties of Continuous Functions. M. K. Fort, Jr. The American Mathematical Monthly, Vol. 59, No. 6 (Jun. - Jul., 1952), pp. 372-375. http://www.jstor.org/stable/2306806.

[Edit: 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:

Park, S. (1999). Ninety years of the Brouwer fixed point theorem. Vietnam Journal of Mathematics, 27(3), 187-222. http://www.math.ac.vn/publications/vjm/vjm_27/No.3/187-222_Park.PDF

And the reference under discussion is:

W. Kulpa, An integral criterion for coincidence property, Radovi Mat.6 (1990) 313-321.

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.  If anyone can find an accessible copy of Kulpa's paper, I would be most interested in it (and I know he would be as well).

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:

Some Properties of Continuous Functions. M. K. Fort, Jr. The American Mathematical Monthly, Vol. 59, No. 6 (Jun. - Jul., 1952), pp. 372-375. http://www.jstor.org/stable/2306806.

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:

Park, S. (1999). Ninety years of the Brouwer fixed point theorem. Vietnam Journal of Mathematics, 27(3), 187-222. http://www.math.ac.vn/publications/vjm/vjm_27/No.3/187-222_Park.PDF.

And the reference under discussion is:

W. Kulpa, An integral criterion for coincidence property, Radovi Mat.6 (1990) 313-321.

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.

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:

Some Properties of Continuous Functions. M. K. Fort, Jr. The American Mathematical Monthly, Vol. 59, No. 6 (Jun. - Jul., 1952), pp. 372-375. http://www.jstor.org/stable/2306806.

[Edit: 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:

Park, S. (1999). Ninety years of the Brouwer fixed point theorem. Vietnam Journal of Mathematics, 27(3), 187-222. http://www.math.ac.vn/publications/vjm/vjm_27/No.3/187-222_Park.PDF

And the reference under discussion is:

W. Kulpa, An integral criterion for coincidence property, Radovi Mat.6 (1990) 313-321.

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. If anyone can find an accessible copy of this paper, I would be most interested in it (and I know he would be as well).

I believe the answer to this question is actually yes, and would be quite pleased if someone could communicate a copy of the last reference I have listed here. (My about-page has an email address.)

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:

Some Properties of Continuous Functions. M. K. Fort, Jr. The American Mathematical Monthly, Vol. 59, No. 6 (Jun. - Jul., 1952), pp. 372-375. http://www.jstor.org/stable/2306806.

[Edit: 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:

Park, S. (1999). Ninety years of the Brouwer fixed point theorem. Vietnam Journal of Mathematics, 27(3), 187-222. http://www.math.ac.vn/publications/vjm/vjm_27/No.3/187-222_Park.PDF

And the reference under discussion is:

W. Kulpa, An integral criterion for coincidence property, Radovi Mat.6 (1990) 313-321.

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. If anyone can find an accessible copy of Kulpa's paper, I would be most interested in it (and I know he would be as well).