I am writing a short survey of some "nice'' properties of algebraic extensions of $\mathbb{Q}$. Let's say a property (P) is nice if

- every finite extension of $\mathbb{Q}$ satisfies (P), and
- if $K \subseteq L \subseteq \overline{\mathbb{Q}}$ and $L$ satisfies (P), then $K$ satisfies (P).

For the property to be interesting, there should be some example of an infinite extension of $\mathbb{Q}$ that does NOT satisfy it. Edit: and there should be an example of an infinite extension that DOES satisfy it!

Here are some examples, described in terms of an algebraic extension $K/\mathbb{Q}$:

-$K^\times/K^\times_{tors}$ is free abelian. (Or replace $\mathbb{G}_m$ with something else (e.g. abelian variety) and ask the same question).

-The ring of integers in $K$ is a Noetherian (and hence a Dedekind domain), or more generally a Lasker ring. These properties are discussed at length in chapter 12 of Ribenboim's book, *The Theory of Classical Valuations*.

-The Northcott and Bogomolov properties (defined in Bombieri and Zannier's paper *A note on heights in certain infinite extensions of $\mathbb{Q}$*). These properties have analogous statements in terms of heights on abelian varieties, etc.

-Conditions on the completions of $K$ at various places, e.g. $K_v$ is a local field for some place $v$, for all $v$, for all $v$ above a fixed rational prime... The case where $K_v$ is a local field with uniformly bounded local degrees is described in detail by Checcoli, *Fields of algebraic numbers with bounded local degrees and their properties*.

If you can think of other properties that fit in this list, please give one per answer, ideally with an explanation of why it is "nice", if it's not obvious.