Odd degree number fields with odd class number

Abstract

For every odd integer $n \geq 3$, we prove that there exist infinitely many number fields of degree $n$ and associated Galois group $S_n$ whose class number is odd. To do so, we study the class groups of families of number fields of degree $n$ whose rings of integers arise as the coordinate rings of the subschemes of $\mathbb{P}^1$ cut out by integral binary $n$-ic forms. By obtaining upper bounds on the mean number of $2$-torsion elements in the class groups of fields in these families, we prove that a positive proportion (tending to $1$ as $n$ tends to $\infty$) of such fields have trivial $2$-torsion subgroup in their class groups and narrow class groups. Conditional on a tail estimate, we also prove the corresponding lower bounds and obtain the exact values of these averages, which are consistent with the heuristics of Cohen-Lenstra-Martinet-Malle and Dummit-Voight. Additionally, for any order $\mathcal{O}_f$ of degree $n$ arising from an integral binary $n$-ic form $f$, we compare the sizes of $\mathrm{Cl}_2(\mathcal{O}_f)$, the $2$-torsion subgroup of ideal classes in $\mathcal{O}_f$, and $\mathcal{I}_2(\mathcal{O}_f)$, the $2$-torsion subgroup of ideals in $\mathcal{O}_f$. For the family of orders arising from integral binary $n$-ic forms and contained in fields with fixed signature $(r_1,r_2)$, we prove that the mean value of the difference $|\mathrm{Cl}_2(\mathcal{O}_f)| - {2^{1-r_1-r_2}}|\mathcal{I}_2(\mathcal{O}_f)|$ is equal to $1$, generalizing a result of Bhargava and the third-named author for cubic fields. Conditional on certain tail estimates, we also prove that the mean value of $|\mathrm{Cl}_2(\mathcal{O}_f)| - {2^{1-r_1-r_2}}|\mathcal{I}_2(\mathcal{O}_f)|$ remains $1$ for certain families obtained by imposing local splitting and maximality conditions.