Multiply monogenic orders

Bérczes, Attila; Evertse, Jan-Hendrik; Győry, Kálmán

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 2, pp. 467-497.

Let $A=\mathbb{Z}[x_1,...,x_r]\supset \mathbb{Z}$ be a domain which is finitely generated over $\mathbb{Z}$ 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 $𝒪\supset A$ with quotient field $K$ which is integral over $A$. $A$-orders in $K$ of the type $A\left[\alpha \right]$ are called monogenic. It was proved by Győry [10] that for any given $A$-order $𝒪$ in $K$ there are at most finitely many $A$-equivalence classes of $\alpha \in 𝒪$ with $A\left[\alpha \right]=𝒪$, where two elements $\alpha ,\beta$ of $𝒪$ are called $A$-equivalent if $\beta =u\alpha +a$ for some $u\in A^{*}$, $a\in A$. If the number of $A$-equivalence classes of $\alpha$ with $A\left[\alpha \right]=𝒪$ is at least $k$, we call $𝒪$ $k$ times monogenic.

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 $\ge 3$, 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.

Classification : 11R99, 11D99, 11J99

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