Counting dihedral and quaternionic extensions

Abstract:

We give asymptotic formulas for the number of biquadratic extensions of $\mathbb {Q}$ that admit a quadratic extension which is a Galois extension of $\mathbb {Q}$ with a prescribed Galois group, for example, with a Galois group isomorphic to the quaternionic group. Our approach is based on a combination of the theory of quadratic equations with some analytic tools such as the Siegel–Walfisz theorem and the double oscillations theorem.