Dirichlet series associated to quartic fields with given resolvent

TL;DR

This paper derives an explicit formula for the Dirichlet series summing over quartic fields with a fixed cubic resolvent, extending previous work on cubic fields with quadratic resolvent.

Contribution

It provides a complete, explicit formula for the Dirichlet series of quartic fields with a specified cubic resolvent, including detailed proofs to replace reliance on unpublished results.

Findings

Explicit Dirichlet series formula for quartic fields with given cubic resolvent

Complete proofs of previous unpublished results included

Extension of methods from quadratic to cubic resolvent cases

Abstract

Let $k$ be a cubic field. We give an explicit formula for the Dirichlet series $\sum_K|\Disc(K)|^{-s}$, where the sum is over isomorphism classes of all quartic fields whose cubic resolvent field is isomorphic to $k$. Our work is a sequel to an unpublished preprint of Cohen, Diaz y Diaz, and Olivier, and we include complete proofs of their results so as not to rely on unpublished work. This is a companion to a previous paper where we compute the Dirichlet series associated to cubic fields having a given quadratic resolvent.