Number of unramified extension of $ \Bbb{Q}_p$ of given degree is $1$, correct ?
I may made some misunderstanding.Îs my following argument correct?
$ \Bbb {Q}_p$($p$ is odd prime) has $p＋1$ extensions from local class field theory, and there is only one unramified extension because there is bijection between 'unramified extension of local field of degree $n$' and 'Corresponding extension of residue field of degree $n$'.But for given $n$, there is only one degree $n$ extension of residue field. So, I conclude there is only one unramified extension of degree $n$, and other $n-1$ extensions ramify(may totally ramify, someone not).