Abstract. PBW deformations of Artin-Schelter regular algebras are skew Calabi-Yau. We prove that the Nakayama automorphisms of such PBW deformations can be obtained from their homogenizations. Some Calabi-Yau properties are generalized without Koszul assumption. We also show that the Nakayama automorphisms of such PBW deformations control Hopf actions on them.
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 PGL2(Q). Following this procedure and working in the group PGL2(Q(\/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 §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" -tic number fields. However, to select a subfamily so that the fields become Galois over Q is not easy and still a lot of work on these remains to be done.
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