Let $K$ be a finite extension of $\mathbb{Q}_p$, and $K^{\text{nr}}$ be the maximal unramified extension of $K$, then I want to ask for an arbitary integer $n$, does there always exist a unique tamely ramified extension of $K^{\text{nr}}$ of degree $n$? If it exists uniquely, must it be a cyclic extension?
This question is motivated by a proof on the bottom of page 359 in this paper of Alain Kraus.
Thanks!
Best Answer
$K^{ur}= \bigcup_{p\ \nmid\ m} K(\zeta_m)$. Its integers are a DVR with residue field $\overline{\Bbb{F}}_p$.
Thus a finite extension $L/K^{ur}$ is totally ramified of degree $n$, its uniformizer satisfies $ \pi_L^d = u \pi_{K^{ur}},u\in O_L^\times$, since $O_L = \sum_{j=0}^{m-1} \pi_L^j O_{K^{ur}}$ we have $d=n$, here $\pi_{K^{ur}}=p$ and $u = \zeta_m (1+\pi_L a),a\in O_L$,
Tamely ramified means $p \nmid n$, from the binomial series $(1+\pi_L a)^{1/n} \in L$ which means $u^{1/n} \in L$ so that $\varpi_L= p^{1/n}$ is an uniformizer and $L = K^{ur}(p^{1/n})$.