Let us fix a number field $K$ of degree $d$ and a (highly non-canonical) isomorphism $\iota:\Bbb{C}\cong \widehat{\overline{\Bbb{Q}_p}}$.
- An archimedean place is an embedding $K\hookrightarrow \Bbb{C} $ up by conjugation in $Gal(\mathbb{C}/\Bbb{R})$
- An non-archimedean place is an embedding $K\hookrightarrow \overline{\Bbb{Q}_p} $ up to conjugation by $Gal(\overline{\mathbb{Q}_p}/\Bbb{Q}_p)$ (see also this answer)
Basically, I would like to know, if I can compare these two somehow via the isomorphism $\iota$. Certainly there is a map
$$ \iota_*:\{\text{embeddings }K\hookrightarrow \overline{\Bbb{Q}_p}\}\longrightarrow \{\text{embeddings }K\hookrightarrow \mathbb{C}\}$$
sending an embedding $K\hookrightarrow \overline{\Bbb{Q}_p}$ to the composition $K\hookrightarrow \overline{\Bbb{Q}_p}\rightarrow \widehat{\overline{\Bbb{Q}_p}}\rightarrow{} \mathbb{C}$. If I am not mistaken, this is an isomorphism, but it shouldn't descend to the respective sets of places, since on the left hand side we have a much bigger group action. But I can ask for something weaker/different:
Question: Say I would like to send one non-archimedean embedding $a$ to an archimedean embedding $b$ of my choice via $\iota_*$. Can I find such an $\iota$? Is $\iota_*$ then uniquely determined?