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In Chapter 3 of the book ''A Course in $p$-adic Analysis'' A.M Robert defines the field $\Omega_p$. He calls the field ''universal'' but doesn't show a universal property. I would have guessed that it's the universal spherical complete and algebraically complete extension of ${\mathbb Q}_p$, but the book doesn't claim anything of the sort, at least I didn't find it. The construction of the field depends on the choice of an ultrafilter and there's not even a remark on the (in)dependence of that choice.

What I mean by universal property is this: I expect something of the following form: Let $Z$ be a field extension of ${\mathbb Q}_p$ with an absolute value extending the one of ${\mathbb Q}_p$ such that $Z$ is algebraically closed and spherically complete. Then there exists an isometric field homomorphism $\Omega_p\to Z$.

Now my question is, whether $\Omega_p$ is indeed universal in the described sense and where I would find a proof of that fact.

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    $\begingroup$ Some people use "universal" to refer to an object such that any other object of a particular sort embeds into or is a quotient of it (e.g. the Cantor set has a property like this). Note that this is not a universal property in the categorical sense. $\endgroup$
    – Qiaochu Yuan
    Oct 5, 2015 at 16:19
  • $\begingroup$ @Qiaochu Yuan: I would be content, if there was a proof of such a statement in this particular case. $\endgroup$
    – user1688
    Oct 5, 2015 at 16:28
  • $\begingroup$ I've used Robert's book a lot, and I think he is using the word "universal" in the sense that this is the biggest extension of $\mathbb{Q}_p$ he is going to use in what follows. Like when you fix a "universal set" in naive set theory. $\endgroup$
    – efs
    Jul 11, 2021 at 18:46

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You need the concept of maximal completeness.

In a word, $\Omega_p$ is the minimal spherically complete field containing $\mathbb{C}_p$ and is the maximal mixed-characteristic valued field with redisue field $\bar{\mathbb{F}}_p$ and valuation group $\mathbb{Q}$.

For details, see this survey.

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In the bad old days when algebraic geometry discourse followed Weil's "Foundations," everything lived within an algebraically closed field with enormous transcendence degree.

It was referred to as "the" universal domain, selected once for all.

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