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Power integral bases for real cyclotomic fields

Published online by Cambridge University Press:  17 April 2009

Laurel Miller-Sims  and
Leanne Robertson
Laurel Miller-Sims
Affiliation:
Department of Mathematics, Smith College, Northampton, MA 01063, United States of America, e-mail: millerlg@math.mcmaster.ca, lroberts@math.smith.edu
Leanne Robertson
Affiliation:
Department of Mathematics, Smith College, Northampton, MA 01063, United States of America, e-mail: millerlg@math.mcmaster.ca, lroberts@math.smith.edu
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We consider the problem of determining all power integral bases for the maximal real subfield Q (ζ + ζ−1) of the p-th cyclotomic field Q (ζ), where p ≥ 5 is prime and ζ is a primitive p-th root of unity. The ring of integers is Z[ζ+ζ−1] so a power integral basis always exists, and there are further non-obvious generators for the ring. Specifically, we prove that if or one of the Galois conjugates of these five algebraic integers. Up to integer translation and multiplication by −1, there are no additional generators for p ≤ 11, and it is plausible that there are no additional generators for p > 13 as well. For p = 13 there is an additional generator, but we show that it does not generalise to an additional generator for 13 < p < 1000.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 1 , February 2005 , pp. 167 - 173
Copyright
Copyright © Australian Mathematical Society 2005

References

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