Background:

This question is related to a proof offered by Gian Maria Dall'Ara.

In Ways to prove the fundamental theorem of algebra (FTA), he formulates the FTA as:

Every complex non-constant polynomial $p$ is surjective.

Dall'Ara's proof may be outlined as follows:

1. Let $C$ be the finite set of critical points, i.e., such that $p′(z)=0$. Note $C$ is finite by elementary algebra.

2. Put $B := \mathbb{C} \backslash p(C)$, and $A := p^{-1}(B)$. Thus $A$ is open, being the complement of a finite subset of $\mathbb{C}$.

3. If $p'(z_0) \neq 0$, then it is elementary that $p$ takes small open neighborhoods of $z_0$ to open neighborhoods of $p(z_0)$. Thus $p$ defines an open map from $A$ to $B$. But also $p: A \to B$ is closed, because any polynomial mapping is proper. In particular, $p(A)$ is both open and closed in $B$; since $B$ is connected, it follows that $p: A \to B$ is surjective, whence $p: \mathbb{C} \to \mathbb{C}$ is also surjective.

It seems to me that one could embellish this proof so as to prove the fundamental theorem of algebra as a consequence of Brouwer's fixed point theorem. Hence my question:

Is there already in the literature a proof of the fundamental theorem of algebra as a consequence of Brouwer's fixed point theorem?