ray class field of rational function field - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:23:30Z http://mathoverflow.net/feeds/question/17305 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17305/ray-class-field-of-rational-function-field ray class field of rational function field norondion 2010-03-06T18:35:38Z 2010-03-06T19:00:01Z <p>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?</p> http://mathoverflow.net/questions/17305/ray-class-field-of-rational-function-field/17308#17308 Answer by Felipe Voloch for ray class field of rational function field Felipe Voloch 2010-03-06T19:00:01Z 2010-03-06T19:00:01Z <p>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</p> <p>$\phi_{T^{n+1}} = \phi_T(\phi_{T^n})$ and $\phi_T =X^q+TX$. </p> <p>It even has the bonus of being an Eisenstein polynomial at $f$.</p>