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An algorithmic construction of cyclic \(p\)-extensions of fields, with characteristic different from \(p\), not containing the \(p\)th roots of unity. (English) Zbl 1247.12008

From Kummer theory we know how to build the cyclic \(p\)-extensions of fields \(E/F\), provided that \(F\) contains \(\zeta_{p^n}\), the \(p^n\)th roots of unity, where \(p^n=[E:F]\). There also some other well known results like Albert’s theorem, which states that \(E/F\) can be embedded into a cyclic \(p\)-extension \(K/F\) of degree \(p^{n+1}\) if and only if \(\zeta_p=\zeta_{p^n}^{p^{n-1}}\) is contained in the image of the norm map \(N_{E/F}\).
In [R. Massy, Acta Arith. 103, No. 1, 21–26 (2002; Zbl 1006.11060)] the author gave an algorithmic construction of any cyclic \(p\)-extension of fields with characteristic different from \(p\), containing only the \(p\)th roots of unity.
In the paper under review Massy improves his previous results by removing the requirement for the presence of a \(p\)th root of unity in the base field \(F\). The existence of \(p\)-cyclic extensions is determined by a similar condition as in Albert’s theorem: \(\zeta_p\) must be contained in the image of the norm map \(N_{E(\zeta_p)/F(\zeta_p)}\). In his main result the author gives an algorithmic computable primitive element for any cyclic \(p\)-extension. The construction involves norm maps, trace forms and \(p\)-Galois averages. As an example, the author constructs explicitly the primitive element for any cyclic \(3\)-extension over \(\mathbb Q_3\), the local field of \(3\)-adic numbers.

MSC:

12F10 Separable extensions, Galois theory
11R20 Other abelian and metabelian extensions
11R32 Galois theory
11S20 Galois theory

Citations:

Zbl 1006.11060
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