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Power bases for cyclotomic integer rings. (English) Zbl 0923.11150

Let \(p\) be an odd prime, \(\xi\) a primitive \(p\)-th root of unity and \({\mathcal O}_p\) the ring of integers of the cyclotomic field \(\mathbb{Q}(\xi)\). It is a classical problem to investigate power integral bases in \(\mathbb{Q}(\xi)\), that is to determine all \(\alpha\) such that \({\mathcal O}_p=\mathbb{Z}[\alpha]\) or equivalently \(\{ 1,\alpha,\ldots,\alpha^{p-1}\}\) is an integral basis. A. Bremner [J. Number Theory 28, 288-298 (1988; Zbl 0637.12001)] conjectured that, up to translations and conjugation only \(\alpha=\xi,1/(1+\xi)\) generate power integral bases. Bremner proved this conjecture for \(p=7\). The author establishes a criterion for Bremner’s conjecture for a given regular prime \(p\). The proof of this criterion is based on a determinant formula for the relative class number of \(\mathbb{Q}(\xi)\). This criterion is then used to verify the conjecture for \(p\leq 23\), \(p\neq 17\).
Reviewer: I.Gaál (Debrecen)

MSC:

11R23 Iwasawa theory
11R18 Cyclotomic extensions
11R04 Algebraic numbers; rings of algebraic integers

Citations:

Zbl 0637.12001
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References:

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