Counting biquadratic number fields with quaternionic and dihedral extensions

TL;DR

This paper derives asymptotic formulas for counting biquadratic number fields with specific Galois group embeddings, using Hilbert symbols and Heath-Brown's method to analyze quadratic character sums.

Contribution

It provides new asymptotic counts for biquadratic fields embedded in quaternionic or dihedral extensions, connecting inverse Galois problems with analytic number theory techniques.

Findings

Derived asymptotic formulas for the number of such fields

Expressed solvability conditions via Hilbert symbols

Applied Heath-Brown's method to bound character sums

Abstract

We establish asymptotic formulae for the number of biquadratic number fields of bounded discriminant that can be embedded into a quaternionic or a dihedral extension. To prove these results, we express the solvability of these inverse Galois problems in terms of Hilbert symbols, and then apply a method of Heath-Brown to bound sums of linked quadratic characters.