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

Abstract

We consider the totally real cyclic quintic fields K n = Q ( ϑ n ) K_{n}=\mathbb {Q}(\vartheta _{n}) , generated by a root ϑ n \vartheta _{n} of the polynomial f n ( x ) = x 5 + n 2 x 4 − ( 2 n 3 + 6 n 2 + 10 n + 10 ) x 3   + ( n 4 + 5 n 3 + 11 n 2 + 15 n + 5 ) x 2 + ( n 3 + 4 n 2 + 10 n + 10 ) x + 1. \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 + 5 n 3 + 15 n 2 + 25 n + 25 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 K_{n} and prove that K n K_{n} has a power integral basis only for n = − 1 , − 2 n=-1,-2 . For n = − 1 , − 2 n=-1,-2 (both values presenting the same field) all generators of power integral bases are computed.