Normal integral bases for Emma Lehmer's parametric family of cyclic quintics (Q558193)

The parametric family of totally real cyclic quintic fields \(L_n\) of \textit{E. Lehmer} [Math. Comput. 50, No. 182, 535--541 (1988; Zbl 0652.12004)] was investigated several authors e.g. by \textit{R. Schoof} and \textit{L. C. Washington} [Math. Comput. 50, No. 182, 543--556 (1988; Zbl 0649.12007)]. \textit{I. Gaál} and \textit{M. Pohst} [Math. Comput. 66, No. 220, 1689--1696 (1997; Zbl 0899.11064)] gave an integral basis in \(L_n\) in a parametric form and used it to study power integral bases of \(L_n\) of the form \(\{1,\alpha,\alpha^2,\alpha^3,\alpha^4\}\). In the present paper the authors give necessary and sufficient condition for the existence of a normal integral basis of the form \(\{v+w\rho,v+w\rho',v+w\rho'',v+w\rho''',v+w\rho''''\}\) in \(L_n\), where \(\rho\) is a root of the polynomial defining \(L_n\), \(v,w\in Z\).