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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Unit groups and class numbers of real cyclic octic fields


Author: Yuan Yuan Shen
Journal: Trans. Amer. Math. Soc. 326 (1991), 179-209
MSC: Primary 11R20; Secondary 11R27, 11R29
MathSciNet review: 1031243
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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 $ \S5$, 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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1031243-3
PII: S 0002-9947(1991)1031243-3
Keywords: Simplest octic fields, class number, unit group
Article copyright: © Copyright 1991 American Mathematical Society