On real cyclic sextic fields

Abstract:

A table of units and class numbers of real cyclic sextic fields with conductor ${f_6} \leqslant 2021$ has been given by the second author [13]. We first fill in the gaps in [13] and then construct an extended table for $2021 < {f_6} < 4000$. The article contains results about Galois module structure of the unit group, relative norms of the units, and ideal classes of the subfields becoming principal in the sextic field. The connection with Leopoldt’s theory [11] is described. A parametric family of fields containing exceptional units [14] is constructed. We give statistics referring to class numbers of fields with prime conductor, the appearance of units of different types if the relative class number is $> 1$, Leopoldt’s unit index, and the signature rank of the unit group.