Multiply monogenic orders (Q2844281)

Let \(A=\mathbb Z[x_1,\dots,x_r]\) be a domain finitely generated over \(\mathbb Z\) and integrally closed in its quotient field \(L\). Let \(L\) be a finite extension field of \(L\). \(A\)-orders in \(K\) of the type \(A[\alpha]\) are called monogenic.NEWLINENEWLINEThe problem of monogeneity and power integral bases is a classical topic in algebraic number theory.NEWLINENEWLINE In this paper, this problem is extended to a more general setting which might also have some applications.NEWLINENEWLINEFor a given \(A\)-order \(O\) the elements \(\alpha,\beta\in O\) are called \(A\)-equivalent if \(\beta=u\alpha+a\) for some \(u\) unit in \(A\) and \(a\in A\). If there are \(k\) equivalence classes of \(\alpha\in O\) with \(A[\alpha]=O\), then we call \(O\) \(k\)-times monogenic.NEWLINENEWLINEThe authors prove that if \(K\) is of degree at least three, then there are only finitely many 3-times monogenic A-orders in \(K\).NEWLINENEWLINEThe authors prove that there are extensions \(K\) which have infinitely many of special 2-times monogenic orders. Under certain conditions on the Galois groups it is shown that \(K\) has only finitely many 2-times monogenic \(A\)-orders which are not of these special types.NEWLINENEWLINESome applications to canonical number systems are given.