MULTIPLY MONOGENIC ORDERS

Abstract

Let A = Z(x1;:::;xr) Z be a domain which is nitely gen- erated over Z and integrally closed in its quotient eld L. Further, let K be a nite extension eld of L. An A-order in K is a domainO A with quotient eld K which is integral overA. A-orders inK of the typeA( ) are called monogenic. It was proved by Gy} ory (10) that for any given A-order O in K there are at most nitely many A-equivalence classes of 2 O with A( ) = O, where two elements ; of O are called A-equivalent if = u +a for some u2 A , a2 A. If the number of A-equivalence classes of with A( ) =O is at least k, we callO k times monogenic. In this paper we study orders which are more than one time monogenic.