The Hasse Norm Principle For Biquadratic Extensions

We give an asymptotic formula for the number of biquadratic extensions of the rationals of bounded discriminant that fail the Hasse norm principle.Comment: 19 pages. Proof of Theorem 1 improved/simplified. Accepted by Journal de Th\'eorie des Nombres de Bordeau

Enumerative Galois theory for cubics and quartics

We show that there are $O_\varepsilon(H^{1.5+\varepsilon})$ monic, cubic polynomials with integer coefficients bounded by $H$ in absolute value whose Galois group is $A_3$. We also show that the order of magnitude for $D_4$ quartics is $H^2 (\log H)^2$, and that the respective counts for $A_4$, $V_4$, $C_4$ are $O(H^{2.91})$, $O(H^2 \log H)$, $O(H^2 \log H)$. Our work establishes that irreducible non-$S_3$ cubic polynomials are less numerous than reducible ones, and similarly in the quartic setting: these are the first two solved cases of a 1936 conjecture made by van der Waerden