# Examples of “nice” properties of algebraic extensions of $\mathbb{Q}$

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

1. every finite extension of $\mathbb{Q}$ satisfies (P), and
2. 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.

For $E$ a fixed elliptic curve (or more generally, abelian variety) over $\mathbb Q$, the condition "$E(K)$ is a finitely generated abelian group" is an answer to your question. It is satisfied by any finite extension $K$ of $\mathbb Q$, by the theorem of Mordell-Weil. If $K \subset L$ and the condition is satisfied by $L$, then it is also satisfied by $K$ since $E(K)$ is a subgroup of $E(L)$ and a subgroup of a finitely generated abelian group is still finitely generated.
The condition is definitely not satisfied for $K=\overline{ \mathbb Q}$. But a theorem of Ribet (combined with results of Kato, Rohrlich and Greenberg) states that this condition is satisfied for $K$ the maximal abelian extension of $\mathbb Q$ unramified outside a fixed set of primes. For other infinite extensions of $\mathbb Q$, the situation is very complex (certainly not fully understood even conjecturally), but there are a good deal of deep conjectures, and a good lot of them proved, concerning extensions of number fields whose Galois group is an analytic pro-$p$-group: this is the so-called "Iwasawa theory for elliptic curve". Of course, the abelian Iwasawa theory, where we deal with abelian such extension, that is of Galois group $\mathbb Z_p^n$, is more advanced, but by no means complete. For a thorough discussion of that theory, see Greenberg's Introduction to Iwasawa Theory for Elliptic Curves.