A property of ramification groups and numbers of algebraic finite extensions of valued fields (Q1098888)

\textit{M. Krasner} [Mem., Cl. Sci., Coll. Quarto, Ser. II, Acad. R. Belg. 11, No.8, (1937)] proved that \((n_ q-n_{q+1})v_ q\equiv 0 (mod h)\), where e is the ramification order of the normal extension K/k, p is the characteristic of the residual field, h is the greatest factor of e prime to p, \(v_ q\) is the q-th ramification number and \(n_ q\) is the order of the q-th ramification group of K/k. - J.-P. Serre extended this result to the classical case of discrete valuations and residual extension \(\bar K/\bar k\) of \(K/k\) separable. The author proves a theorem describing some congruences which concern orders of q-th ramification groups \(V_ q\) and orders of some special overgroups \(V'_ q\) of \(V_{q+1}\) for dense valuations of a field k and arbitrary normal extension \(K/k.\) In the classical case these congruences lead to the above mentioned results.