On computing discriminants of subfields of \(\mathbb Q(\zeta_{p^r})\)
Let \(p\) be an odd prime, \(r\) a positive integer and \(K\) a subfield of the \(p^r\)th cyclotomic field \(\mathbb Q(\zeta_{p^r})\). Then the absolute degree of \(K\) is of the form \(up^j\), where \(u\) is a divisor of \(p-1\) and \(j\) is an integer satisfying \(0\leq j\leq r-1\). Due to the well-known cyclicity of the Galois group of \(\mathbb Q(\zeta_{p^r})/\mathbb Q\), its absolute degree fully characterizes \(K\) among the subfields of \(\mathbb Q(\zeta_{p^r})\). The main result of the paper consists in the computation of the discriminant of \(K\) via Hasse conductor-discriminant formula. Is is shown that the absolute value of the discriminant of \(K\) is equal to \(p^a\), where \(a=u((j+2)p^j-(p^{j+1}-1)/(p-1))-1\).
scientific article; zbMATH DE number 1886721
scientific article