# Counting Cubic Extensions with given Quadratic Resolvent

2 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : Given a number field $k$ and a quadratic extension $K_2$, we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of $k$ whose Galois closure contains $K_2$ as quadratic subextension, ordered by the norm of their relative discriminant ideal. The main tool is Kummer theory. We also study in detail the error term of the asymptotics and show that it is $O(X^{\alpha})$, for an explicit $\alpha<1$.
Document type :
Journal articles

https://hal.archives-ouvertes.fr/hal-00463533
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Submitted on : Friday, March 12, 2010 - 3:43:07 PM
Last modification on : Thursday, January 20, 2022 - 5:31:38 PM

### Citation

Henri Cohen, Anna Morra. Counting Cubic Extensions with given Quadratic Resolvent. Journal of Algebra, Elsevier, 2011, 325, pp.461-478. ⟨10.1016/j.jalgebra.2010.08.027⟩. ⟨hal-00463533⟩

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