Absolute norms of \(p\)-primary units. (English) Zbl 1214.11131

Let \(K\) be a finite extension of the \(p\)-adic field \({\mathbb Q}_p\) which contains a primitive \(p^m\) root of unity for some \(m\geq 1\). Say that the unit \(\alpha\in K\) is \(p\)-primary if \(K(\alpha^{1/p})/K\) is an unramified extension. In this paper it is proved that if \(\alpha\) is a \(p\)-primary unit in \(K\) then \(\text{N}_{K/{\mathbb Q}_p}(\alpha)\equiv 1\pmod{p^{m+1}}\).


11S15 Ramification and extension theory
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