Multiply monogenic orders

Details Multiply monogenic orders Multiply monogenic orders

2011-03-24

arxiv.org/abs/1103.4740v1

Mathematics

Number Theory

40 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Multiply monogenic orders does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Multiply monogenic orders, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Multiply monogenic orders will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-47580