Yes you can prove FTA using BFPT. See also recent paper by Daniel Reem relating to Open Mapping Theorem.
I often looked at the 1949 note of B.H. Arnold (a professor of mine in bygone times) but missed the flaw. Maybe someone could post the essence of the retraction by him and Ivan Niven. Thank you.
You have proved FTA when you prove that any irreducible *real polynomial has degree 1 or two.
The existence of an *irreducible one of higher degree leads to a real vector space of that same dimension on which polynomial product induces a ring structure, in fact, integral domain.
Consider the squaring map on this real algebra, and scale it to map the whole thing to a Cartesian sphere of dimension one less. Scaling the domain vector variable also, we get a squaring mapping from unit sphere to unit sphere, in fact from RP(n-1) to the sphere. The mapping is topologically continuous (on Hausdorff spaces) due to bilinearity of the original product operation.
The integrity condition (entire domain) shows that this mapping is injective. Everything in sight is compact Hausdorff, so such a 1-1 mapping induces a homeomorphism to the image.
Throw in "connectedness" and the (Brouwer) Invariance of Domain Theorem shows that in fact the image of RP(n-1) must be the whole sphere. The fact that these two spaces are thus homeomorphic leads to the conclusion that n<=2 .
To show that RP is not topologically the same as S(n-1), construct a non-trivial loop in RP by projecting a longitude from North to South pole on its universal covering sphere.
This loop could not be homotopic to the constant loop (at the chosen base point) since the homotopy would lift to the UC keeping end-points invariant.
What does this have to do with BFPT? As Dr. Terence Tao points out in his blog, the latter theorem is necessary in some form to prove Invariance of Domain.