Counting dihedral and quaternionic extensions
Étienne Fouvry, Florian Luca, Francesco Pappalardi, Igor E. Shparlinski · Transactions of the American Mathematical Society · 2011
We give asymptotic formulas for the number of biquadratic extensions of Q \mathbb {Q} that admit a quadratic extension which is a Galois extension of Q \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.