Odd degree number fields with odd class number

Abstract

For every odd integer $n\ge 3$, 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 subgroups 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 and Lenstra, Cohen and Martinet, Malle, and Dummit and 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 ${Cl}_2\left({\mathcal{O}}_f\right)$, the $2$-torsion subgroup of ideal classes in ${\mathcal{O}}_f$, and of ${\mathcal{I}}_2\left({\mathcal{O}}_f\right)$, 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 $\left|{Cl}_2\right({\mathcal{O}}_f\left)\right|-2^{1-r_1-r_2}\left|{\mathcal{I}}_2\right({\mathcal{O}}_f\left)\right|$ 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 $\left|{Cl}_2\right({\mathcal{O}}_f\left)\right|-2^{1-r_1-r_2}\left|{\mathcal{I}}_2\right({\mathcal{O}}_f\left)\right|$ remains $1$ for certain families obtained by imposing local splitting and maximality conditions.