Multiply monogenic orders
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 2, pp. 467-497.

Let A=[x 1 ,...,x r ] be a domain which is finitely generated over and integrally closed in its quotient field L. Further, let K be a finite extension field of L. An A-order in K is a domain 𝒪A with quotient field K which is integral over A. A-orders in K of the type A[α] are called monogenic. It was proved by Győry [10] that for any given A-order 𝒪 in K there are at most finitely many A-equivalence classes of α𝒪 with A[α]=𝒪, where two elements α,β of 𝒪 are called A-equivalent if β=uα+a for some uA * , aA. If the number of A-equivalence classes of α with A[α]=𝒪 is at least k, we call 𝒪 k times monogenic.

In this paper we study orders which are more than one time monogenic. Our first main result is that if K is any finite extension of L of degree 3, then there are only finitely many three times monogenic A-orders in K. Next, we define two special types of two times monogenic A-orders, and show that there are extensions K which have infinitely many orders of these types. Then under certain conditions imposed on the Galois group of the normal closure of K over L, we prove that K 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.

Publié le :
Classification : 11R99, 11D99, 11J99
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     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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Bérczes, Attila; Evertse, Jan-Hendrik; Győry, Kálmán. Multiply monogenic orders. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 2, pp. 467-497. http://archive.numdam.org/item/ASNSP_2013_5_12_2_467_0/

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