Normal integral bases of Lehmer's cyclic quintic fields

TL;DR

This paper characterizes all normal integral bases of Lehmer's cyclic quintic fields generated by roots of a specific polynomial, extending previous results to more general cases based on the polynomial's prime or square-free nature.

Contribution

It provides a complete description of normal integral bases for these fields using roots of the polynomial, generalizing earlier work for special cases.

Findings

All normal integral bases are given explicitly by the roots of the polynomial.

The results extend Lehmer's and Spearman-Williams' work to broader classes of fields.

The characterization depends on the primality or square-freeness of a polynomial expression.

Abstract

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