Dirichlet series associated to quartic fields with given cubic resolvent

Henri Cohen (1, 2) , Frank Thorne (3)

1 LFANT - Lithe and fast algorithmic number theory
2 IMB - Institut de Mathématiques de Bordeaux
3 Department of Mathematics [Columbia]

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 Our work is a sequel to an unpublished preprint of Cohen, Diaz y Diaz, and Olivier, and we include complete proofs of their results so as not to rely on unpublished work.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.

Domaines: Théorie des nombres [math.NT]

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