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

Authors:	Istvá n Gaá l Michael Pohst

Institution:	Kossuth Lajos University, Mathematical Institute, H--4010 Debrecen Pf.12., Hungary ; Technische Universität Berlin, Fachbereich 3 Mathematik, Straße des 17. Juni 136, 10623 Germany

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

Keywords:	Power integral basis family of quintic fields

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