Biquadratic fields that embed into a quaternionic or dihedral extension have density zero and are counted by explicit asymptotics of order sqrt(X) (log X)^(1/2).
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Counting biquadratic number fields with quaternionic and dihedral extensions
Biquadratic fields that embed into a quaternionic or dihedral extension have density zero and are counted by explicit asymptotics of order sqrt(X) (log X)^(1/2).