Counting Cubic Extensions with given Quadratic Resolvent

Details Counting Cubic Extensions with given Quadratic Resolvent Counting Cubic Extensions with given Quadratic Resolvent

2010-03-09

arxiv.org/abs/1003.1869v1

Journal of Algebra 325 (2011), pp. 461-478

Mathematics

Number Theory

19 pages

Scientific paper

10.1016/j.jalgebra.2010.08.027

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 $\alpha<1$.

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