×

A note on normal and power bases. (English) Zbl 0632.12010

Let \(K/\mathbb Q\) be a cyclic extension of prime degree \(p\) and \(\varepsilon\) an algebraic unit in \(K\). Let \(\ell\) be any prime number for which \(\mathbb Q_{\ell}(\varepsilon)\neq \mathbb Q_{\ell}\). The author shows the following implication: If \(\varepsilon\) is a normal integral generator of \(K\) over \(\mathbb Q\), then \(\varepsilon\) generates the ring of integers in \(\mathbb Q_{\ell}(\varepsilon)\) as algebra over \(\mathbb Z_{\ell}\). For further results see M.-N. Gras [J. Number Theory 23, 347–353 (1986; Zbl 0564.12008)].
Reviewer: R.Massy

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11R20 Other abelian and metabelian extensions
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers

Citations:

Zbl 0564.12008

References:

[1] CASSELS J. W. S., FRÖHLICH. A.: Algebraic Number Theory. Russian edition, Izdateľstvo ”MIR”, Moscow 1969. · Zbl 0153.07403
[2] HASSE H.: Number Theory. Akademie-Verlag-Berlin 1979. · Zbl 0423.12001
[3] NARKIEWICZ W.: Elementary and Analytic Theory of Algebraic Numbers. PWN, Warszawa 1974. · Zbl 0276.12002
[4] VAN DER WAERDEN B. L.: Algebra. Russian edition, Izdateľstvo ”Nauka”, Moscow 1976. · Zbl 0997.00501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.