We show that there are order of magnitude $H^2 (\log H)^2$ monic quartic polynomials with integer coefficients having box height at most $H$ whose Galois group is $D_4$. Further, we prove that the corresponding number of $V_4$ and $C_4$ quartics is $O(H^2 \log H)$. Finally, we show that the count for $A_4$ quartics is $O(H^{2.95})$. Our work establishes that irreducible non-$S_4$ quartics are less numerous than reducible quartics. 查看全文>>