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Given any family of cubic fields defined by local conditions at finitely manyprimes, we determine the mean number of 2-torsion elements in the class groupsand narrow class groups of these cubic fields when ordered by their absolutediscriminants.  For an order $\cal O$ in a cubic field, we study the three groups: $\rmCl_2(\cal O)$, the group of ideal classes of $\cal O$ of order 2; $\rmCl^+_2(\cal O)$, the group of narrow ideal classes of $\cal O$ of order 2; and${\cal I}_2(\cal O)$, the group of ideals of $\cal O$ of order 2. We prove thatthe mean value of the difference $|\rm Cl_2({\cal O})|-\frac14|{\cal I}_2(\calO)|$ is always equal to $1$, whether one averages over the maximal orders inreal cubic fields, over all orders in real cubic fields, or indeed over anyfamily of real cubic orders defined by local conditions. For the narrow classgroup, we prove that the mean value of the difference $|\rm Cl^+_2({\calO})|-|{\cal I}_2(\cal O)|$ is equal to $1$ for any such family. For any familyof complex cubic orders defined by local conditions, we prove similarly thatthe mean value of the difference $|\rm Cl_2(\mathcal O)|-\frac12|{\calI}_2(\cal O)|$ is always equal to $1$, independent of the family.  The determination of these mean numbers allows us to prove a number offurther results as by-products. Most notably, we prove---in stark contrast tothe case of quadratic fields---that: 1) a positive proportion of cubic fieldshave odd class number; 2) a positive proportion of real cubic fields haveisomorphic 2-torsion in the class group and the narrow class group; and 3) apositive proportion of real cubic fields contain units of mixed real signature.We also show that a positive proportion of real cubic fields have narrow classgroup strictly larger than the class group, and thus a positive proportion ofreal cubic fields do not possess units of every possible real signature.

On the mean number of 2-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields