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

Pisolkar, Supriya

J. Théor. Nombres Bordx. 21, No. 3, 733-740 (2009).

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

Reviewer: Kevin Keating (Gainesville FL)

MSC:

11S15

Ramification and extension theory

Keywords:

p-primary units; Hasse-Herbrand functions

References:

[1]	Dalawat C.S., Local discriminants, kummerian extensions, and abelian curves. ArXiv:0711.3878. · Zbl 1226.11118
[2]	Fesenko, I. B., Vostokov, S. V., Local fields and their extensions. Second edition. Translations of Mathematical Monographs 121, AMS Providence, RI, 2001. · Zbl 1156.11046
[3]	Hasse H., Number theory. Classics in Mathematics. Springer-Verlag, Berlin, 2002. · Zbl 0991.11001
[4]	Martinet, J., Les discriminants quadratiques et la congruence de Stickelberger. Sém. Théor. Nombres Bordeaux (2) 1 (1989), no. 1, 197-204. · Zbl 0731.11061
[5]	Neukirch, J., Class field theory. Grund. der Math. Wiss. 280. Springer-Verlag, Berlin, 1986. · Zbl 0587.12001
[6]	Jean-Pierre Serre, Local Fields. GTM 67, Springer-Verlag, 1979. · Zbl 0423.12016