### 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 |
---|---|

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 - 2014 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Mathematics (miscellaneous)

### Cite this

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

*12*(2), 467-497.

**Multiply monogenic orders.** / Bérczes, Attila; Evertse, Jan Hendrik; Györy, K.

Research output: Contribution to journal › Article

*Annali della Scuola normale superiore di Pisa - Classe di scienze*, vol. 12, no. 2, pp. 467-497.

}

TY - JOUR

T1 - Multiply monogenic orders

AU - Bérczes, Attila

AU - Evertse, Jan Hendrik

AU - Györy, K.

PY - 2014

Y1 - 2014

N2 - Let A = Z[x1,., xr] ⊃ 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.

AB - Let A = Z[x1,., xr] ⊃ 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.

UR - http://www.scopus.com/inward/record.url?scp=84908451267&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908451267&partnerID=8YFLogxK

M3 - Article

VL - 12

SP - 467

EP - 497

JO - Annali della Scuola Normale - Classe di Scienze

JF - Annali della Scuola Normale - Classe di Scienze

SN - 0391-173X

IS - 2

ER -