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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.

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    How about starting with a *statement* of the FTA in the language of +,×,0,1 ... ??? Such a statement, if it exists, would belong to the **theory**, at least in the way I understand that word.2012-01-07
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    @ismythe: What _is_ the first-order theory of the complex field? As you know, no first-order theory with infinite models can be absolutely categorical...2012-01-07
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    You are right, there is an issue with my wording, as the full statement of the FTA requires a natural number quantifier. The next best thing would be to consider, for each $n$, the sentence (taking $z,x_0,x_1,\ldots$ as my list of variables in the usual language of fields): $\forall x_0\forall x_1\cdots\forall x_n \exists z (x_nz^n+x_{n-1}z^{n-1}+\cdots+x_1z+x_0 = 0)$2012-01-07
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    Also, see my edit.2012-01-07

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