Mathematics of Computation

Author(s): István Gaál; Michael Pohst.
Journal: Math. Comp. 66 (1997), 1689-1696.
MSC (1991): Primary 11Y50; Secondary 11Y40, 11D57
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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

(both values presenting the same field) all generators of power integral bases are computed.

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Retrieve articles in Mathematics of Computation with MSC (1991): 11Y50, 11Y40, 11D57

István Gaál
Affiliation: Kossuth Lajos University, Mathematical Institute, H--4010 Debrecen Pf.12., Hungary
Email: igaal@math.klte.hu

Michael Pohst
Affiliation: Technische Universität Berlin, Fachbereich 3 Mathematik, Straße des 17. Juni 136, 10623 Germany
Email: pohst@math.tu-berlin.de

DOI: 10.1090/S0025-5718-97-00868-5
PII: S 0025-5718(97)00868-5
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
Copyright of article: Copyright 1997, American Mathematical Society

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