On relations for rings generated by algebraic numbers and their conjugates

Abstract

Let \(\alpha \) be an algebraic number of degree \(d\) with minimal polynomial \(F \in \mathbb {Z}[X]\), and let \(\mathbb {Z}[\alpha ]\) be the ring generated by \(\alpha \) over \(\mathbb {Z}\). We are interested whether a given number \(\beta \in \mathbb {Q}(\alpha )\) belongs to the ring \(\mathbb {Z}[\alpha ]\) or not. We give a practical computational algorithm to answer this question. Furthermore, we prove that a rational number \(r/t \in \mathbb {Q}\), where \(r \in \mathbb {Z}, t \in \mathbb {N}, \gcd (r, t) = 1\), belongs to the ring \(\mathbb {Z}[\alpha ]\) if and only if the square-free part of its denominator \(t\) divides all the coefficients of the minimal polynomial \(F \in \mathbb {Z}[X]\) except for the constant coefficient \(F(0)\) that must be relatively prime to \(t\), namely \(\gcd (F(0),t)=1\). We also study the question when the equality \(\mathbb {Z}[\alpha ] = \mathbb {Z}[\alpha ']\) for algebraic numbers \(\alpha , \alpha '\) conjugates over \(\mathbb {Q}\) holds. In particular, it is shown that for each \(d \in \mathbb {N}\), there are conjugate algebraic numbers \(\alpha , \alpha '\) of degree \(d\) satisfying \(\mathbb {Q}(\alpha ) = \mathbb {Q}(\alpha ')\) and \(\mathbb {Z}[\alpha ] \ne \mathbb {Z}[\alpha ']\). The question concerning the equality \(\mathbb {Z}[\alpha ]=\mathbb {Z}[\alpha ']\) is answered completely for conjugate quadratic pairs \(\alpha ,\alpha '\) and also for conjugate pairs \(\alpha , \alpha '\) of cubic algebraic integers.