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

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.