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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Unit groups and class numbers of real cyclic octic fields
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by Yuan Yuan Shen PDF
Trans. Amer. Math. Soc. 326 (1991), 179-209 Request permission

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.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 326 (1991), 179-209
  • MSC: Primary 11R20; Secondary 11R27, 11R29
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1031243-3
  • MathSciNet review: 1031243