Constructions of power bases of cyclotomic fields
For an algebraic number field \(K\) with ring of integers \(O_ K\), an algebraic integer \(\alpha\) is said to generate a power basis of \(K\) if \(\mathbb{Z}[\alpha]=O_ K\). For the \(m\)th cyclotomic field \(\mathbb{Q}(\zeta_ m)\) (\(\zeta_ m\) primitive \(m\)th root of unity), it is shown in this paper that both \(\alpha_ m=\zeta_ m+\zeta_ m^ 2+\dots+\zeta_ m^{(m-1)/2}\) (if \(m\) is odd) and \(\beta_ m=\prod_{1<j<m, (j,m)=1} (1-\zeta_ m^ j)\) (if \(m\) has at least 2 distinct prime divisors) generate power bases. The author also conjectures that there are no other power basis generators of \(\mathbb{Q}(\zeta_ m)\) except the obvious one \(\zeta_ m\), up to equivalence \(\sim\) viz \(\alpha\sim\beta\) if \(\alpha=k\pm \sigma(\beta)\), \(k\in\mathbb{Z}\), \(\sigma\in \text{Gal}(K/\mathbb{Q})\) and proves that this is indeed the case for \(m=8,9,12\).
scientific article; zbMATH DE number 149410