Timeline for Extension of $\mathbb C_p$?

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Jul 8, 2019 at 13:41	comment	added	reuns		Your question is weird because ($\Bbb{Q}_p$ being complete discrete valuation field) $v$ extends in a unique way to finite extensions of $\Bbb{Q}_p$ thus to $\overline{\Bbb{Q}}_p$ and there is a unique continuous (continuity of $a \mapsto 2^{-v(a)}$) extension to $\Bbb{C}_p$. So everything is about the normalization, the valuation you chose for $p$ (or $\pi_v$). $K_v$ is a finite extension of $\Bbb{Q}_p$, if $K/\Bbb{Q}$ is Galois then $K_v$ depends only on $K,p$, if it is not Galois it may depend on the prime of $O_K$ (the elements with valuation $> 0$) above $p$ you chose.
Jul 8, 2019 at 12:30	comment	added	joaopa		What do you mean by "compatibly with the valuations"? Do you mean that $v_1(f(x))=v_p(x)$ for all $x\in\mathbb C_p$ where $f$ is an isomorphism of $\mathbb C_p$ into $\mathbb D_v$?
Jul 8, 2019 at 12:16	comment	added	SashaP		An algebraic closure of $K_v$ is isomorphic to $\bar{\mathbb{Q}_p}$ compatibly with the valuations because $K_v$ is a finite extension of $\mathbb{Q}_p$, so $D_v$ is isomorphic to $\mathbb{C}_p$.
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Jul 8, 2019 at 9:46	history	asked	joaopa	CC BY-SA 4.0