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Given a field and a metric on it, consider the goal of completing it and extending it in order to get an algebraicly closed and complete field.

How should one proceed? Should one first complete it and then extend it, or the other way around? Why?

Example: in $\mathbb{Q}$ with the absolute value metric, the best way to go is to complete it to $\mathbb{R}$ and then extend it to $\mathbb{C}$. If we do it in the other way, one ends up infinitely extending $\mathbb{Q}$ with non-algebraic reals. Is it expected that completing first is the best strategy?

Example: in $\mathbb{Q}$ with the $p$-adic metric, one complete it to $\mathbb{Q}_p$, extend it infinitely many times to $\bar{\mathbb{Q}}_p$ and then complete it again to $\Omega$. Is this the best strategy?

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  • $\begingroup$ For $p$-adic numbers, one can do it with a single completion : first choose an algebraic closure $\bar{\mathbb{Q}}$, let $\bar{\mathbb{Z}}$ be the integral closure of $\mathbb{Z}$ in it, consider $A$ the $p$-adic completion of the local ring of $\bar{\mathbb{Z}}$ at some (chosen) maximal ideal above $p$, and finally let $\Omega = A [ \frac{1}{p}]$. $\endgroup$
    – js21
    Commented Apr 20, 2017 at 16:48
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    $\begingroup$ The completion of an algebraically closed valued field is algebraically closed. So, for any valued field, just two steps are enough: first take algebraic closure, then take completion. $\endgroup$
    – Emil Jeřábek
    Commented Apr 20, 2017 at 20:51
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    $\begingroup$ What metric are you applying, Shake, to decide which of two strategies is the better one? $\endgroup$
    – Gerry Myerson
    Commented Apr 20, 2017 at 23:20

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