Counting Cubic Extensions with given Quadratic Resolvent

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 $\alpha <1$.

19 pages

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• Identities for field extensions generalizing the Ohno-Nakagawa relations
• Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent
• Four Perspectives on Secondary Terms in the Davenport-Heilbronn Theorems
• Dirichlet series associated to quartic fields with given resolvent
• Shintani's zeta function is not a finite sum of Euler products
• The shapes of pure cubic fields