Normal integral bases of Lehmer’s cyclic quintic fields

Let \(K_n\) $K_n$ be a tamely ramified cyclic quintic field generated by a root of Emma Lehmer’s parametric polynomial. We give all normal integral bases for \(K_n\) $K_n$ only by the roots of the polynomial, which is a generalization of the work of Lehmer in the case that \(n^4+5n^3+15n^2+25n+25\) $n^4+5n^3+15n^2+25n+25$ is prime number, and Spearman–Willliams in the case that \(n^4+5n^3+15n^2+25n+25\) $n^4+5n^3+15n^2+25n+25$ is square free.