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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
##### Keywords:
polynomial cycle; normal basis; ambiguous ideal