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It is a well-known fact, proved in every introductory textbook on algebraic number theory, that if $K$ is an algebraic number field, i.e. a finite extension of $\mathbb{Q}$, then its ring $\mathcal{O}_K$ of integers is a free abelian group.

Does this statement still hold for arbitrary algebraic extensions of $\mathbb{Q}$? In particular, is the underlying abelian group of the ring $\mathcal{O}_{\overline{\mathbb{Q}}}$ of all algebraic integers free abelian?

Should this be true, I am also interested whether anything is known about the dependence of this statement on the axiom of choice, and similar logical questions.

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up vote 29 down vote accepted

Pontryagin's criterion says that, for a countable, torsion-free, abelian group to be free, it suffices that every finitely many elements lie in a finitely generated pure subgroup. The rings $\mathcal O_K$ for finite extensions $K$ of $\mathbb Q$ show, in view of the result you quoted, that this criterion is satisfied. The axiom of choice is not needed for any of this, because the question is absolute between the full universe $V$ and Gödel's constructible universe $L$.

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That's great, thank you! – Robert Kucharczyk Dec 22 '11 at 19:47
I think you have misquoted Pontryagin's criterion. I'm not sure what the word "pure" means, so I am not positive, but it sounds like the criterion you are describing would apply to the additive group of ℚ which is, of course, not free. In Robinson's group theory book, the condition is on finite rank subgroups, not merely finitely generated subgroups – David Speyer Dec 22 '11 at 19:56
@David: Among torsion-free abelian groups, a subgroup $H$ of a group $G$ is said to be pure iff the quotient $G/H$ is torsion-free; equivalently, whenever $H$ contains a non-zero multiple of an element $x$ of $G$, it also contains $x$ itself. In particular, the only pure subgroups of $\mathbb Q$ are $\mathbb Q$ itself and 0. Also, any finite-rank subgroup is included in the purification of a finite subset $F$ (i.e., the smallest pure subgroup that includes $F$). – Andreas Blass Dec 22 '11 at 20:05
I think there is also the terminology "saturated" for such subgroups. – François Brunault Dec 23 '11 at 18:06
Motivated by Francois Brunault's comment, I googled "saturated subgroup" and found an impressive number of different meanings for this phrase (or at least they look different to me). And they didn't look like the model-theoretic meaning of "saturated" either. An old name for pure subgroups was "serving subgroups," a translation of Prüfer's "Servanzuntergruppe". – Andreas Blass Dec 23 '11 at 19:27

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