Dirichlet series associated to quartic fields with given cubic resolvent

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 Our work is a sequel to an unpublished preprint of Cohen, Diaz y Diaz, and Olivier, and we include complete proofs of their results so as not to rely on unpublished work.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.
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Submitted on : Tuesday, August 27, 2013 - 5:49:12 PM
Last modification on : Thursday, January 11, 2018 - 6:22:36 AM

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

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