Dirichlet series associated to quartic fields with given cubic resolvent

Let k be a cubic field. We give an explicit formula for the Dirichlet series ∑ 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 Our work is a sequel to the unpublished preprint [ 12 ] whose results have been summa...

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Bibliographic Details
Published inResearch in number theory Vol. 2; no. 1; pp. 1 - 40
Main Authors Cohen, Henri, Thorne, Frank
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.12.2016
Springer Nature B.V
Springer
Subjects
Dirichlet problem
Isomorphism
Mathematics
Mathematics and Statistics
Number Theory
Research Article
Unpublished Preprint
Cyclic Cubic
Trivial Norm
Quartic Field
Splitting Types
Online AccessGet full text
ISSN2363-9555
2363-9555
DOI10.1007/s40993-015-0001-y

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Summary:Let k be a cubic field. We give an explicit formula for the Dirichlet series ∑ 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 Our work is a sequel to the unpublished preprint [ 12 ] whose results have been summarized in [ 7 ], so we include complete proofs so as not to rely on unpublished work. This is a companion paper to [ 14 ] where we compute the Dirichlet series associated to cubic fields having a given quadratic resolvent. Mathematics Subject Classification: 11R16
Bibliography:ObjectType-Article-1
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content type line 14
ISSN:2363-9555
2363-9555
DOI:10.1007/s40993-015-0001-y