Unit groups and class numbers of real cyclic octic fields

Abstract:

The generating polynomials of D. Shanks’ simplest quadratic and cubic fields and M.-N. Gras’ simplest quartic and sextic fields can be obtained by working in the group ${\mathbf {PG}}{{\mathbf {L}}_2}({\mathbf {Q}})$. Following this procedure and working in the group ${\mathbf {PG}}{{\mathbf {L}}_2}({\mathbf {Q}}(\sqrt 2))$, we obtain a family of octic polynomials and hence a family of real cyclic octic fields. We find a system of independent units which is close to being a system of fundamental units in the sense that the index has a uniform upper bound. To do this, we use a group theoretic argument along with a method similar to one used by T. W. Cusick to find a lower bound for the regulator and hence an upper bound for the index. Via Brauer-Siegel’s theorem, we can estimate how large the class numbers of our octic fields are. After working out the first three examples in $\S 5$, we make a conjecture that the index is $8$. We succeed in getting a system of fundamental units for the quartic subfield. For the octic field we obtain a set of units which we conjecture to be fundamental. Finally, there is a very natural way to generalize the octic polynomials to get a family of real ${2^n}$-tic number fields. However, to select a subfamily so that the fields become Galois over ${\mathbf {Q}}$ is not easy and still a lot of work on these remains to be done.