The discriminant of abelian number fields (Q521285)

The discriminant is one of the most important invariants of an algebraic number field. For any abelian number field \(K\), \textit{J. C. Interlando} et al. [J. Algebra Appl. 5, No. 1, 35--41 (2006; Zbl 1151.11056)] gave a formula for computing the discriminant of \(K\) involving the conductor \(m\) of \(K\), the degree of \(K\) over the field \(\mathbb{Q}\) of rational numbers and the degrees of the fields \(K\cap \mathbb{Q}(\zeta_{m/p^{\alpha}})\) over \(\mathbb{Q}\), where \(p\) runs through the set of primes that divide \(m\), \(p^{\alpha}\) runs through all powers of \(p\) that divide \(m\). In this paper, the authors find out a similar formula without using the condition over the conductor of \(K\). They also point out the mistake in Theorem 2 of Interlando et al. [loc. cit.] and correct it. At the end of the paper, the authors give nice families of examples to illustrate the main result.