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Mathematics of Computation

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Power integral bases in a parametric family of totally real cyclic quintics

Authors: István Gaál and Michael Pohst
Journal: Math. Comp. 66 (1997), 1689-1696
MSC (1991): Primary 11Y50; Secondary 11Y40, 11D57
Erratum: Math. Comp. 50 (1988), 653.
Erratum: Math. Comp. 41 (1983), 778-779.
MathSciNet review: 1423074
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Abstract: We consider the totally real cyclic quintic fields $K_{n}=\mathbb {Q}(\vartheta _{n})$, generated by a root $\vartheta _{n}$ of the polynomial \begin{multline*} f_{n}(x)=x^{5}+n^{2}x^{4}-(2n^{3}+6n^{2}+10n+10)x^{3}\ +(n^{4}+5n^{3}+11n^{2}+15n+5)x^{2}+(n^{3}+4n^{2}+10n+10)x+1. \end{multline*} Assuming that $m=n^{4}+5n^{3}+15n^{2}+25n+25$ is square free, we compute explicitly an integral basis and a set of fundamental units of $K_{n}$ and prove that $K_{n}$ has a power integral basis only for $n=-1,-2$. For $n=-1,-2$ (both values presenting the same field) all generators of power integral bases are computed.

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Additional Information

István Gaál
Affiliation: Kossuth Lajos University, Mathematical Institute, H–4010 Debrecen Pf.12., Hungary

Michael Pohst
Affiliation: Technische Universität Berlin, Fachbereich 3 Mathematik, Straße des 17. Juni 136, 10623 Germany

Keywords: Power integral basis, family of quintic fields
Received by editor(s): August 13, 1996
Additional Notes: Research supported in part by Grants 16791 and 16975 from the Hungarian National Foundation for Scientific Research and by the Deutsche Forschungsgemeinschaft
Article copyright: © Copyright 1997 American Mathematical Society