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Let $f \in \mathbf{F}_q[T]$ be irreducible. I know that the ray class field for $\mathrm{Cl}((f)) \cong (\mathbf{F}_q[T]/(f))^\times$ can be constructed by adjoining torsion points of a Carlitz module. Is there an easy explicit minimal polynomial for a generator of this extension?

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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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