Multiply monogenic orders

TL;DR

This paper investigates the properties and finiteness of orders in algebraic number fields that are generated by a single element multiple times, revealing finiteness results for three times monogenic orders and conditions for two times monogenic orders.

Contribution

It establishes finiteness results for three times monogenic orders in number fields of degree at least 3 and characterizes certain types of two times monogenic orders, extending previous understanding.

Findings

Finitely many three times monogenic orders in fields of degree ≥ 3.

Existence of infinitely many two times monogenic orders of specific types.

Finiteness of two times monogenic orders outside special types under Galois group conditions.

Abstract

Let O be an order in an algebraic number field K, i.e., a ring with quotient field K which is contained in the ring of integers of K. The order O is called monogenic, if it is of the shape Z[w], i.e., generated over the rational integers by one element. By a result of Gy\H{o}ry (1976), the set of w with Z[w]=O is a union of finitely many equivalence classes, where two elements v,w of O are called equivalent if v+w or v-w is a rational integer. An order O is called k times monogenic if there are at least k different equivalence classes of w with Z[w]=O, and precisely k times monogenic if there are precisely k such equivalence classes. It is known that every quadratic order is precisely one time monogenic, while in number fields of degree larger than 2, there may be non-monogenic orders. In this paper we study orders which are more than one time monogenic. Our first main result is, that in any number field K of degree at least 3 there are only finitely many three times monogenic orders. Next, we define two special types of two times monogenic orders, and show that there are number fields K which have infinitely many orders of these types. Then under certain conditions imposed on the Galois group of the normal closure of K, we prove that K has only finitely many two times monogenic orders which are not of these types. We give some immediate applications to canonical number systems. Further, we prove extensions of our results for domains which are monogenic over a given domain A of characteristic 0 which is finitely generated over Z.