Is it known whether the Fundamental Theorem of Algebra is a theorem or non-theorem of the first-order theory of the complex field (i.e. $\mathbb{C}$ together with $+,\times,0,1$)?
Every proof I've seen uses some topological properties of the plane, but I was wondering if anyone had answered the question of their necessity from this viewpoint.
EDIT: I see now that the way I have worded this question makes it either unanswerable, or trivial (after all, the FTA is a true theorem of the complex field, hence, in the theory). What I think I was originally curious about is the following, motivated by the question of whether there exists a purely algebraic proof of the FTA, can each sentence in the following schema, for each $n$,
$\forall x_0\forall x_1\cdots\forall x_n\exists z (x_nz^n+\cdots+x_1z+x_0=0)$
be given first-order proofs in some reasonable extension of the first-order axiomatization of fields, which would somehow characterize $\mathbb{C}$, however, there are obvious obstacles to the second part of this statement (via Loewenheim-Skolem Theorems). I will have to rethink my curiosity. Thanks.