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Let $K$ be a number field and $L$ an infinite algebraic extension of $K$. Fix a non-trivial absolute value $v$ on $K$ (so $v$ is induced either by an embedding into the complex numbers or by a prime ideal in the ring of integers of $K$). If $K_v$ is the corresponding completion and $\overline{K}_v$ a choice of algebraic closure of $K_v$, then the extensions of $v$ to an absolute value on $L$ are in bijection with the $Gal(\overline{K}_v/K_v)$-orbits of $Hom_{K-alg}(L,\overline{K}_v)$ (this is described in detail in, e.g., Neukirch's Algebraic Number Theory).

My question regards the case when $v$ is non-Archimedean, arising from a prime ideal $\mathfrak{p}$ with residue characteristic $p$. In this case, is there a bijection between prime ideals of $\mathscr{O}_L$ lying above $\mathfrak{p}$ and places of $L$ above $v$? Since $\mathscr{O}_L$ is not generally going to be Dedekind (though it is a one-dimensional, integrally closed domain), we don't get an (additive) discrete valuation in the usual way from a prime ideal of $\mathscr{O}_L$, but if $w$ is a non-Archimedean absolute value on $L$ extending $v$, then by restricting to finite sub-extensions $L_i$ of $L/K$ we get sort of a coherent sequence of prime ideals $\mathfrak{p}_i$ in the integer rings of the $L_i$. Perhaps there is a way to convert this sequence of primes into a single prime ideal of $\mathscr{O}_L$ lying above each of the $\mathfrak{p}_i$? I thought maybe some kind of compactness\inverse limit type argument might work, but I'm not sure anymore...even so, if such an argument were to work, it would probably just show the existence of such a prime, as opposed to determining it uniquely. Alternatively, if I look at the maximal ideal of the valuation ring of $w$ in $L$ and intersect it with $\mathscr{O}_L$, I should get a (hopefully) non-zero prime ideal of $\mathscr{O}_L$ that maybe fits the bill.

This might be the wrong approach entirely (and maybe the answer to my question is just "no"). The reason I'm interested is because, for example, in Washington's book on cyclotomic fields, he (in an appendix) defines the decomposition group of a prime ideal in an infinite Galois extension of number fields, but makes no mention of the decomposition group of a place of an infinite algebraic extension of a number field. When one starts considering objects like the maximal unramified abelian $p$-extension of a $\mathbb{Z}_p$-extension of a number field, surely this (possibly more general) notion becomes relevant.

I would greatly appreciate if anyone could set me straight on this issue or point me in the direction of a reference where it's discussed. Thanks.

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up vote 10 down vote accepted

Since $\mathcal O_L$ is the union of $\mathcal O_{L_i}$, where $L_i$ runs over all the finite subextensions, giving a prime ideal $\mathfrak p$ in $\mathcal O_L$ is the same as giving a compatible collection of primes ideals $\mathfrak p_i$ in each $\mathcal O_{L_i}$. (We set $\mathfrak p_i := \mathfrak p \cap \mathcal O_{L_i}$, and $\mathfrak p = \cup_i \mathfrak p_i$.)

Since $L$ is the union of the $L_i$, giving an absolute value $v$ on $L$ is the same as giving compatible absolute values $v_i$ on the various $L_i$. (Take $v_i$ to be the restriction to $L_i$ of $v$.)

Combining the preceding two remarks, we see that the bijection between primes ideals and non-Archimedean valuations in the case of finite extensions extends to a corresponding bijection in the case of infinite extensions.

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That's perfect. Thanks very much! –  Keenan Kidwell Sep 30 '10 at 12:03
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