Multiply monogenic orders

作者：Berczes Attila^*; Evertse Jan Hendrik; Gyory Kalman

来源：Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 2013, 12(2): 467-497.

Let A = Z[x(1), ... , x(r)] superset of Z be a domain which is finitely generated over Z 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 O superset of A with quotient field K which is integral over A. A-orders in K of the type A [alpha] are called monogenic. It was proved by Gyory [10] that for any given A-order O in K there are at most finitely many A-equivalence classes of alpha is an element of O with A [alpha] = O, where two elements alpha, beta of O are called A-equivalent if beta = u alpha + a for some u is an element of A*, a is an element of A. If the number of A-equivalence classes of alpha with A [alpha] = O is at least k, we call O k times monogenic. %26lt;br%26gt;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 %26gt;= 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.

• 出版日期2013