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If $K$ is a Global Field, then $\mathcal O_K$ is a Dedekind domain which has many nice properties, one of which being that ideals can be uniquely factorised into prime ideals.

But now suppose $K/\mathbb Q$ is an infinite (algebraic) extension, then I believe $\mathcal O_K$ is no longer a Dedekind domain. In this case which "well known" properties of number fields/global fields are lost and which still hold?

For example, say $L/K$ is a finite extension. If $\mathfrak p$ is a prime ideal of $\mathcal O_K$, does it still make sense to talk about the ramification degree of $\mathfrak p$ in $L$, since the localisation $K_{\mathfrak p}$ might not have a discrete valuation? And would identities like $e_{L/K}f_{L/K}=[L:K]$ still make sense?

In a similar vein, what can be said about infinite algebraic extensions of $\mathbb Q_p$?

Yaneda
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    We have several more knowledgable users. But IIRC in Milne's notes on class field theory there is some information about infinite extensions of $\Bbb{Q}p$. An even more vague recollection is that unramifiied parts (so $f{L/K}$) still works, somehow, and can be studied using tools of Galois theory. Sorry about possibly sending you to chase after something that need not be there :-( – Jyrki Lahtonen Sep 16 '24 at 08:24
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    The first issue I see is that there are no good finiteness properties any more: $\mathcal{O}_K$ is (likely) not Noetherian (wasn’t there a question here about that?). You can have phenomena like infinitely many primes above a given ideal, or infinite ramification indices... I think people tend to avoid infinite extensions of $\mathbb{Q}_p$, because they are not complete, but IIRC a lot of Serre’s Local Fields would hold with completions of infinite extensions of $\mathbb{Q}_p$. It may be worth checking out some Iwasawa theory to see what can be said about very ramified situations. – Aphelli Sep 16 '24 at 08:55
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    See https://math.stackexchange.com/questions/976701/overline-mathbbz-is-not-a-dedekind-domain and https://math.stackexchange.com/questions/468966/ramification-in-the-ring-of-all-algebraic-integers and https://math.stackexchange.com/questions/2422227/is-the-integral-closure-of-mathbbz-in-the-algebraic-closure-of-mathbbq/2422230#2422230 – lhf Sep 16 '24 at 14:44

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A comment to the question mentioned that the Noetherian question was raised here. This property can go both ways: a ring of integers in an infinite-degree algebraic extension of $\mathbf Q$. Look at answers here.

An appendix in Washington's book on cyclotomic fields has a brief discussion of primes in number fields of possibly infinite degree over the rationals along with definitions of the ramification index and residue field degree in the infinite-degree case.

When $K/\mathbf Q$ is a finite Galois extension, every element of ${\rm Gal}(K/\mathbf Q)$ is a Frobenius element at infinitely many prime ideals: that's part of the Chebotarev density theorem. When $K/\mathbf Q$ is an infinite-degree Galois extension, most elements of ${\rm Gal}(K/\mathbf Q)$ aren't Frobenius elements but the Frobenius elements are a dense subset using the Krull topology on the Galois group.

KCd
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