Timeline for Topology of multiplication groups of local fields

7 events

when toggle format	what		by	license	comment
Oct 29, 2021 at 5:34	vote	accept	Alice
Oct 29, 2021 at 5:02	answer	added	KConrad		timeline score: 6
Oct 27, 2021 at 13:32	comment	added	F Zaldivar		I think the argument in Neukirch works: The ${\mathbb Z}_p$-submodule $V\subseteq U^{(1)}$ is free of finite rank $d$ (and therefore closed); the isomorphism $V\simeq {\mathbb Z}_p^d$ sends (topological) generators to (topological) generators.
Oct 27, 2021 at 12:44	comment	added	YCor		A compact $d$-dimensional abelian $p$-adic Lie group is always isomorphic to $\mathbf{Z}_p^d\times F$ for some finite abelian group $F$. Once this is granted, things boil down to understand the torsion part. I assume $U^{(1)}$, which can be written $\mathbf{Z}_p^d\times F'$ with $d=[K:\mathbf{Q}_p]$, is by definition the kernel of $O_K^\times\to F_q^*$. An element in the kernel has the form $1+m$ with $m$ in the maximal ideal. One sees easily using the binomial formula that $(1+m)^p$ tends to zero. So it already follows that $F'$ is a $p$-group.
Oct 27, 2021 at 12:37	history	edited	YCor		edited tags
S Oct 27, 2021 at 12:21	review	First questions
Oct 27, 2021 at 15:11
S Oct 27, 2021 at 12:21	history	asked	Alice	CC BY-SA 4.0