### Abstract

Let A = Z[x_{1},., x_{r}] ⊃ ℤ be a domain which is finitely generated over ℤ and integrally closed in its quotient field L. Further, let AT be a finite extension field of L. An A-order in κ is a domain O⊃ A with quotient field κ which is integral over A. ,4-orders in κ of the type A[α] are called monogenic. It was proved by Gyóry [10] that for any given A-order O in κ there are at most finitely many A-equivalence classes of α ∈ O with A[α] = O, where two elements α, β of O are called A-equivalent if β = uα + a for some u∈ A ∗, a ∈ A. If the number of A-equivalence classes of a with A[α] = O is at least κ, we call O κ times monogenic. In this paper we study orders which are more than one time monogenic. Our first main result is that if κ is any finite extension of L of degree ≥ 3, then there are only finitely many three times monogenic A-orders in κ. Next we define two special types of two times monogenic A-orders, and show that there are extensions κ which have infinitely many orders of these types. Then under certain conditions imposed on the Galois group of the normal closure of κ over L, we prove that κ has only finitely many two times monogenic A-orders which are not of these types. Some immediate applications to canonical number systems are also mentioned.

Original language | English |
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Pages (from-to) | 467-497 |

Number of pages | 31 |

Journal | Annali della Scuola normale superiore di Pisa - Classe di scienze |

Volume | 12 |

Issue number | 2 |

Publication status | Published - Jan 1 2014 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Mathematics (miscellaneous)

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## Cite this

*Annali della Scuola normale superiore di Pisa - Classe di scienze*,

*12*(2), 467-497.