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 Notherian (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.

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.

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.

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 Notherian (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.

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 Notherian (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.