The minimal polynomial is $phi_f(X)/X$, where $phi_g$ (the Carlitz module) is defined by being $mathbb{F}_q$-linear in $g$, satisfy
$phi_{T^{n+1}} = phi_T(phi_{T^n})$ and $phi_T =X^q+TX$.
It even has the bonus of being an Eisenstein polynomial at $f$.
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