Stickelberger's congruences for absolute norms of relative discriminants (Q628839)

Let \(L/K\) be a finite extension of number fields, unramified at \(2\), \({\mathfrak d}_{L/K}\) the discriminant, \(c\) the number of complex places of \(L\) which lie above a real place of \(K\), \(\overline{N}_{K/{\mathbb Q}}({\mathfrak d}_{L/K})\) the absolute norm of \({\mathfrak d}_{L/K}\), \(k\) the maximal subfield of \( {\mathbb Q}(\mu_{2^\infty})\) contained in \(K\), \([k:{\mathbb Q}]=2^m,m\geq 0\). The author proves \[ (-1)^c\overline{N}_{K/{\mathbb Q}}({\mathfrak d}_{L/K})\equiv 1\pmod {4\cdot 2^m}. \]