×

zbMATH — the first resource for mathematics

On orbits in ambiguous ideals. (English) Zbl 1017.11012
Let \(K\) be a tamely ramified cyclic algebraic number field of prime degree \(\ell\) over \(\mathbb Q\). The author gives a connection between the existence of a \(\mathbb Z\)-basis \((\alpha_0,\dots,\alpha_\ell)\) for an ambiguous ideal \(\mathcal F\) of the ring of integers of \(K\) and the existence of a polynomial \(f\in \mathbb Z[X]\) such that for \(i=0,\dots ,\ell-2\), one has \(f(\alpha_i)=\alpha_{i+1}\), \(f(\alpha_{\ell-1})=\alpha_0\). In case \(f\) exists, the author constructs a “power” basis of \(\mathcal F\).

MSC:
11C08 Polynomials in number theory
12F05 Algebraic field extensions
PDF BibTeX XML Cite