# Dirichlet series associated to quartic fields with given cubic resolvent

1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
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 Our work is a sequel to an unpublished preprint of Cohen, Diaz y Diaz, and Olivier, and we include complete proofs of their results so as not to rely on unpublished work.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.
Document type :
Journal articles

https://hal.inria.fr/hal-00854664
Contributor : Andreas Enge Connect in order to contact the contributor
Submitted on : Tuesday, August 27, 2013 - 5:49:12 PM
Last modification on : Thursday, January 20, 2022 - 5:31:36 PM

### Citation

Henri Cohen, Frank Thorne. Dirichlet series associated to quartic fields with given cubic resolvent. Research in Number Theory, Springer, 2016, 2 (29), pp.1-40. ⟨10.1007/s40993-015-0001-y⟩. ⟨hal-00854664⟩

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