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It is perhaps instructive to see how Todd's proof of the FTA steps outside of $\mathsf{RCA}_0$ and how it could be modified to fit into $\mathsf{RCA}_0$.

It is perhaps instructive to see how Todd's proof of the FTA steps outside of $\mathsf{RCA}_0$ and how it could be modified to fit into $\mathsf{RCA}_0$.

Todd first step uses the Bolzano-Weierstrass Theorem (BWT) for closed discs in $\mathbb{C}$. At first sight, this requires $\mathsf{ACA}_0$ but the point is to show that $|f(z)|$ attains a minimum value. The weaker subsystem $\mathsf{WKL}_0$ already proves that every real-valued function on a totally bounded complete metric space attains a minimum value [Simpson Theorem IV.2.2].

Steps 2 and 3 of Todd's proof are computations, so now we have a proof in $\mathsf{WKL}_0$. Since the Extreme Value Theorem is equivalent to $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$ [Simpson Theorem IV.2.3] it seems that we can't do much better. This would be the case if $f(z)$ were an arbitrary continuous function, but polynomials aren't arbitrary at all!

Todd first step uses the Bolzano-Weierstrass Theorem (BWT) for closed discs in $\mathbb{C}$. At first sight, this requires $\mathsf{ACA}_0$ but the point is to show that $|f(z)|$ attains a minimum value. The weaker subsystem $\mathsf{WKL}_0$ already proves that every continuous real-valued function on a totally bounded complete metric space attains a minimum value [Simpson Theorem IV.2.2]. The remaining parts of Todd's proof are direct computations, so now we have a proof in $\mathsf{WKL}_0$. Since the Extreme Value Theorem is equivalent to $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$ [Simpson Theorem IV.2.3] it seems that we can't do much better. This would be the case if $f(z)$ were an arbitrary continuous function, but polynomials aren't arbitrary at all!

Now back to Todd's proof. As in Todd's proof, let $f(z)$ be a nonconstant polynomial over $\mathbb{C}$.

$\mathsf{WKL}_0$ is strictly stronger than $\mathsf{RCA}_0$ and $\mathsf{ACA}_0$ is strictly stronger than $\mathsf{WKL}_0$. The first-order fragment of $\mathsf{ACA}_0$ is $\mathsf{PA}$ and every model of $\mathsf{PA}$ can be expanded to a model of $\mathsf{ACA}_0$; the $\mathsf{RCA}_0$ and $\mathsf{WKL}_0$ have the same first-order fragment, $\mathsf{PA}$ with induction restricted to $\Sigma_1$-formulas, and likewise any first-order model of that theory can be expanded to a model of $\mathsf{WKL}_0$ (and hence of $\mathsf{RCA}_0$).

Now back to Todd's proof. As in Todd's proof, let $f(z)$ be a nonconstant polynomial over $\mathbb{C}$.