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?



ray class field of rational function field
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 $f$.


