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A survey on monogenic orders. (English) Zbl 1249.11102

Summary: Recall that an order \(\mathcal O\) in an algebraic number field \(K\) is called monogenic if it is generated by one element, i.e., there is an \(\alpha\) with \(\mathbb Z[\alpha ]=\mathcal O\).By work of K. Győry [Publ.Math.Debrecen 23, 141–165 (1976; Zbl 0354.10041)] there are, up to a suitable equivalence, only finitely many \(\alpha\) such that \(\mathbb Z[\alpha ]=\mathcal O\). In this survey, we give an overview of recent results on estimates for the number of \(\alpha\) up to equivalence.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11D57 Multiplicative and norm form equations
11-02 Research exposition (monographs, survey articles) pertaining to number theory

Citations:

Zbl 0354.10041
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