This question may be too detailed but perhaps somebody knows the answer: Neukirch proofs in his algebraic number theory book in Chapter IV, Proposition 6.2, that his class field axiom implies that the Tate cohomology groups $H^n(G(L|K),U_L)$ for n=0,-1 vanish for finite unramified extensions $L|K$, where $U_L$ is the group of units. He mentions in the proof that every element $a \in A_L$ can be written as $a = \epsilon \pi_K^m$, where $\epsilon \in U_L$ and $\pi_K$ is a prime element in $A_K$. Why does this work?
I absolutely understand this argument when the image of the valuation just lies in $\mathbb{Z}$! But how does this work for a valuation whose image is $\widehat{\mathbb{Z}}$? Unless $A$ is not a profinite module, I don't know what $\pi_K^m$ is for some general $m \in \widehat{\mathbb{Z}}$. Unfortunately, this must work in this generality for global class field theory.
[Math] Neukirch’s class field axiom and cohomology of units for unramified extension
class-field-theorycohomologynt.number-theorynumber-fields
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In response to Minhyong's request, I am reposting my comments above as an answer:
As James Newton commented, if $L/K$ is unramified, then an irreducible $n$-dimensional representation (over $\mathbb C$) of $Gal(L/K)$ will correspond, in the Langlands paradise, to a cuspidal automorphic representation of $GL_n(\mathbb A_K)$. The cuspidal automorphic representations that arise in this way are sometime (especially in the older literature) called "Galois type".
Thus one can (more or less --- there is the issue of irreducible vs. all reps. which I won't think about here) encode unramified extensions of $K$ whose Galois groups admit $n$-dimensional representations in terms of Galois type cuspidal automorphic representations $\pi$ of $GL_n(\mathbb A_K)$ that are unramified at every finite prime.
Now the question arises: how many such $\pi$ are there, and can one compute them?
Being of Galois type is (conjecturally, but we are in paradise!) purely a condition on $\pi_v$ for primes $v$ of $K$ lying over $\infty$, and in fact there are a finite number of prescribed representations of $GL_n(K_v)$ ($= GL_n(\mathbb R)$ or $GL_n(\mathbb C)$) which are allowed. (E.g. for $GL_n(K)$, the possibilities are limit of discrete series, corresponding to holomorphic weight one forms, or principal series with $\lambda = 1/4$, corresponding to Maass forms with eigenvalue of Laplacian equal to $1/4$.) For a given $n$ and $K$, these can be enumerated.
Now since we are asking that the "weight" (i.e. the collection of $\pi_v$ for $v|\infty$) be bounded (i.e. lie in a given finite set), and we are also asking that "level" be one (i.e. that there is no ramification at any finite prime), there are only a finite set of $\pi$ corresponding to irreducible everywhere unramified $n$-dimensional complex representations of $GL_n(Gal(\bar{K}/K)$. [Aside: Minhyong asked for a sketch of a proof of this; here goes: fixing the representation at infinity means that we are fixing a bunch of elliptic operators that the automorphic forms must satisfy. Fixing the level means that we are working on some particular quotient $X/\Gamma$ (here $X$ is the symmetric space attached to the real group in question, in our particular case $GL_n(K\otimes \mathbb R)$, and $\Gamma$ is the fixed level). This need not be compact (indeed won't be in our particular case), but the cuspidal condition (indeed, even the moderate growth condition that non-cuspidal automorphic forms are required to satisfy) means that we can pretend it is, since we explicitly rule out the possibility of extreme growth at infinity. So now we looking at sections of some bundle on a compact space satsifying a bunch of elliptic equations, and such a space of sections if finite dimensional. (The holomorphic modular forms case is the most familiar: in this case the elliptic equations are the Cauchy--Riemann equations. In the Maass form case, the corresponding fact is the finiteness of the eigenspaces of the Laplacian. These are good models for the general case.)]
To actually compute them (say for a fixed choice of $K$ and $n$) would be quite difficult (as David Hansen notes in his comment). The reason is that the relevant $\pi_v$ for $v|\infty$ are never discrete series (even when $n = 2$, and in any case, note that $GL_n(\mathbb R)$ never has discrete series if $n > 2$, and $GL_n(\mathbb C)$ never has discrete series when $n > 1$), and so standard applications of the trace formula to counting automorphic forms won't work.
Nevertheless, it seems that one might still be able to use the trace formula to analyze the situation, at least in principle. For example, Selberg used his original formulation of the trace formula for $SL_2(\mathbb R)/SL_2(\mathbb Z)$ to compute cuspidal Maass forms of level 1,
and showed that the smallest eigenvalue $\lambda$ that occurs has $\lambda$ much greater than 1/4 (maybe closer to 90?).
And we all know that it is not hard to show that there are no holomorphic weight one forms
of level one. So one can automorphically prove (modulo standard conjectures in the Maass form case) that there are no everywhere unramified two-dimensional complex representations of $Gal(\bar{\mathbb Q}/\mathbb Q)$. (This is of course an incredible battle, even in paradise, for a tiny portion of the information that Minkowski gives us, but is meant just to illustrate that this approach is not a priori ridiculous.)
What I don't see at all from this point of view is how to study all $n$ simultaneously. For example, one could imagine implementing this program and finding, for some $K$ and some $n$, maybe $n = 10^6$, that there are no unramified extensions $L/K$ with $L$ admitting an irrep. of dimension $\leq 10^6$. This doesn't rule out the possibility that there is a beautiful, everywhere unramified extension $L/K$ whose Galois group's lowest degree irrep. happens to be of enormous dimension.
The Langlands program seems to be intrinsically geared to thinking about linear representations of Galois groups, and to set the scene, you have to begin by choosing a linear group, which will then cut everything else down in a Procrustean manner. At least superficially (and this answer reflects just superficial thoughts about the question), it doesn't seem well adapted to questions related to the nature of $\pi_1(\mathcal O)$, where no a priori linear structure is given, or indeed expected.
[Added July 14, in response to Minhyong's question in the comments as to whether or not discrete series can convert into non-discrete series after applying some functoriality. The answer is essentially no, as I will now explain.
Added November 29, 2011: What follows is wrong; the answer seems rather to be yes. (See below for details.)
For an arithmetic geometer, one should think of an automorphic form on the adelic group $G(\mathbb A_K)$ as a morphism from the motivic Galois group (over the base number field $K$) to the $L$-group of $G$. (There are subtleties and caveats, of course, but they need not concern us here; all I will say about them is that automorphic forms can give rise to "motives" with non-integral $(p,q)$ in its Hodge decomposition, which necessitates enlarging the category of motives to an unknown larger category, whose hypothetical Tannakian group is called "the Langlands group".)
Now functoriality takes place when you have a map from the $L$-group of $G$ to the $L$-group of $H$; one can just compose this with a map from the motivic Galois group to the former, to obtain a map from the motivic Galois group to the latter. Functoriality is the assertion that the corresponding automorphic form on $H(\mathbb A_K)$ exists.
Now given an automorphic form $\pi$, its factors at the primes $v|\infty$ encode (via the local Langlands corresondence for $\mathbb R$ or $\mathbb C$) the Hodge numbers of the corresponding motive. One feature of discrete series is that (among other properties) they give rise to regular Hodge numbers, i.e. to sequences of $h^{p,q}$ with each $h^{p,q} \leq 1$. Now our original automorphic rep'n $\pi$ on $G(\mathbb A)$ corresponds to a motive whose Mumford--Tate group lies in the $L$-group of $G$, and if $\pi$ has discrete series components at primes above $\infty$, it has regular Hodge numbers at every place dividing $\infty$. If we then pass to a new motive by applying some map from the $L$-group of $G$ to the $L$-group of $H$, then concretely this corresponds to doing some kind of multilinear algebra on our motive, and the only way this can kill the property of having regular Hodge--Tate weights is if we do something like taking the diagonal map from the $L$-group of $G$ into its product with itself, and then embed the latter into the $L$-group of $H$. All such constructions will necessarily be a "reducible" rep'n of the $L$-group of $G$ in the $L$-group of $H$ (more precisely, the centralizer will be a non-trivial Levi), and the corresponding automorphic form won't be a cuspform.
But even if we destroy the property of having regular Hodge numbers, we typically still don't have an Artin motive. To get an Artin motive we have to have $h^{p,q} = 0$ unless $p = q = 0$, and to do this, we have to do even more destructive things, like map the $L$-group of $G$ into the $L$-group of $H$ via the trivial representation, or something similar. Again, this won't correspond to any kind of interesting automorphic forms, just those that correspond to (certain) sums of characters. So we can't produce interesting Galois type automorphic forms out of automorphic forms whose factors at primes above $\infty$ are discrete series.]
[Correction added Nov. 29, 2011: From the Galois/motivic point of view, we have an algebraic group (the Mumford--Tate group of some motive), with a representation (the particular motive), and the Mumford--Tate group contains a cocharacter whose eigenvalues are the Hodge numbers. Discrete series corresponds to the eigenspaces being one dimensional. We now apply some functoriality, which is essentially to say that we apply some multi-linear algebraic process to the representation. Now this can certainly produce eigenspaces for the cocharacter of multiplicity $> 1$. (E.g. the adjoint representation of $SL_3$ has a two-dimensional eigenspace.) So it seems that functoriality doesn't preserve being discrete series. It does preserve being tempered. And the remarks about not getting Artin motives still seem okay, since while the eigenspaces can become greater than $1$-dimensional, for all the eigenspaces to become trivial, we have to do something pretty destructive, like applying functoriality for the trivial representation.]
Henselian-ness means that any factorisation of $\overline{f}$ into coprime monic polynomials in $\kappa[X]$ lifts to $K[X]$. But the factorisation of $\overline{f}$ could be a power of an irreducible polynomial, and this is indeed what happens in practice: consider $K = \mathbb{Q}_p$ and $L = K[x]$ where $x^2 = p$. Then $\overline{f}$ is just $x^2$, which is obviously reducible but cannot be factorised nontrivially as a product of coprime factors.
Best Answer
You dont need to make sense out of $\pi_K^m$ for a general $m$ in $\hat{\mathbb{Z}}$. All you really need to know for his argument is that $v_K(A_K) = v_L(A_L)$ as subgroups of $\hat{\mathbb{Z}}$. I didn't think this through but I think it should be pretty easy to establish from the fact that $\pi_K$ is prime for both valuations.
All he really uses is that the Galois group fixes $\pi_K$.