When is \(\mathbb{Z}[a]\) the ring of the integers? (Q1239194)

In the paper the following theorem is proved: Let \(R\) be a Dedekind domain and \(\alpha\) an algebraic integer over \(R\). Then \(R[\alpha]\) is a Dedekind domain, if and only if the defining polynomial \(f(x)\) of \(\alpha\) does not belong to the square of any maximal ideal of the polynomial ring \(R[x]\). In particular, this result gives a criterion when \(\mathbb{Q}(\alpha)\), \(\mathbb{Q}\) the field of the rationals, has integral basis of the form \(1,\alpha,\alpha^2,\dots,\alpha^{n-1}\) The author applies his result for giving a short proof of the well-known fact that the cyclotomic fields are of this type.