On norm subgroups of complete discrete valuation fields (Q1901846)

Let \(F\) be a complete field with respect to a discrete valuation with residue field \(\overline {F}\). The author investigates the question for which fields \(F\) a finite abelian extension \(L\) is determined by the subgroup \(N_{L/F} L^\times\) of the multiplicative group \(F^\times\) of \(F\). In general this is not the case. But if \(\overline {F}\) is an intermediate field of \(k((t_1 ))\dots ((t_{n-1} ))/k\), where \(k\) is a perfect field which is not algebraically \(p\)-closed, then the extensions \(L\) can be determined by \(N_{L/K} L^\times\) if \(L\) is totally \(p\)-ramified, where \(p\) is the residue characteristic of \(F\).