## 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 $$\text{N}_{K/{\mathbb Q}_p}(\alpha)\equiv 1\pmod{p^{m+1}}$$.

### MSC:

 11S15 Ramification and extension theory

### Keywords:

p-primary units; Hasse-Herbrand functions
Full Text:

### References:

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