The Brückner-Vostokov formula for the Hilbert symbol and its validity in the case \(p=2\) (Q1911551)

Let \(p\) be a prime, \(\nu \in \mathbb{N}\) and \(K\) a finite extension of \(\mathbb{Q}_p\) which contains the group of \(n=p^\nu\)-th roots of unity. An explicit formula for the \(n\)-th Hilbert symbol for \(p\) odd has been given by \textit{H. Brückner} [Vorlesungen Fachbereich Math. Univ. Essen, Heft 2 (1979; Zbl 0437.12001)] and \textit{S. Vostokov} [Math. USSR, Izv. 13, 557-588 (1979; Zbl 0467.12018)]. \textit{K. Kato} [Bull. Soc. Math. Fr. 119, 397-441 (1991; Zbl 0752.14015)] generalised this result to some regular local rings, by using the crystalline cohomology. The author of the paper under review gives a new proof of the Brückner-Vostokov result and extends it also to the case \(p=2\).