Let A = Z[x1,., xr] ⊃ ℤ be a domain which is finitely generated over ℤ and integrally closed in its quotient field L. Further, let AT be a finite extension field of L. An A-order in κ is a domain O⊃ A with quotient field κ which is integral over A. ,4-orders in κ of the type A[α] are called monogenic. It was proved by Gyóry  that for any given A-order O in κ there are at most finitely many A-equivalence classes of α ∈ O with A[α] = O, where two elements α, β of O are called A-equivalent if β = uα + a for some u∈ A ∗, a ∈ A. If the number of A-equivalence classes of a with A[α] = O is at least κ, we call O κ times monogenic. In this paper we study orders which are more than one time monogenic. Our first main result is that if κ is any finite extension of L of degree ≥ 3, then there are only finitely many three times monogenic A-orders in κ. Next we define two special types of two times monogenic A-orders, and show that there are extensions κ which have infinitely many orders of these types. Then under certain conditions imposed on the Galois group of the normal closure of κ over L, we prove that κ 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.
|Number of pages||31|
|Journal||Annali della Scuola normale superiore di Pisa - Classe di scienze|
|Publication status||Published - Jan 1 2014|
ASJC Scopus subject areas
- Theoretical Computer Science
- Mathematics (miscellaneous)