# Proving the index formula of the norm residue group without $p$-adic completion

I asked the following question in MSE, but I've got no answer so far.

Let $K$ be an algebraic number field. Let $A$ be the ring of integers in $K$. Let $\mathfrak{p}$ be a prime ideal of $K$. Let $A_\mathfrak{p}$ be the localization of $A$ at $\mathfrak{p}$. Let $A_\mathfrak{p}^\times$ be the group of invertible elements of $A_\mathfrak{p}$. Elements of $A_\mathfrak{p}^\times$ are called $\mathfrak{p}$-units.

Let $I$ be a non-zero ideal of $A$. An element $\alpha$ of $A$ is called an $I$-unit if $\alpha$ is a $\mathfrak{p}$-unit for every prime divisor $\mathfrak{p}$ of $I$.

Let $L$ be a finite etension of $K$. Let $B$ be the ring of integers in $L$. Let $\mathfrak{p}$ be a prime ideal of $K$. Let $\lambda > 0$ be an integer. Let $\alpha \in A_\mathfrak{p}^\times$. If there exists a $\mathfrak{p}B$-unit $\Gamma \in L$ such that $\frac{\alpha}{N_{L/K}(\Gamma)} \equiv 1$ (mod $\mathfrak{p}^\lambda A_\mathfrak{p})$, $\alpha$ is called a norm residue of $\mathfrak{p}^\lambda$ with respect to $L/K$. Such $\alpha$ form a subgroup of $A_\mathfrak{p}^\times$. This group is called the norm residue group of $\mathfrak{p}^\lambda$ with respect to $L/K$. We denote this group by $\mathfrak{N}(\mathfrak{p}^\lambda)$.

The following proposition is crucial in the classical proof of the global class field theory. The usual proof uses $\mathfrak{p}$-adic completion.

Proposition Let $L$ be a finite cyclic extension of $K$. Let $\mathfrak{p}$ be a prime ideal of $K$. Let $e$ be the ramification index of $\mathfrak{p}$ in $L$. Then there exists an integer $\lambda_0 > 0$ such that $[A_\mathfrak{p}^\times \colon \mathfrak{N}(\mathfrak{p}^\lambda)] = e$ for all integer $\lambda \ge \lambda_0$.

My question How do we prove this without using the $\mathfrak{p}$-adic completion?

Remark I think Hasse's "Bericht" proves this without using the $\mathfrak{p}$-adic completion. However, my German is not good.

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Whoever has left votes to close, please leave a comment explaining them --- being silent is very rude! Ditto for whoever has voted this down. – Theo Johnson-Freyd Sep 14 '12 at 6:05
Why do you want a global proof? In general, local proofs are easier, and the statement you write is purely local. Have you tried using the very same proof working in the local case also in the global one? By the way, you speak about "usual proof": do you have a reference? – Filippo Alberto Edoardo Sep 14 '12 at 16:33
@FilippoAlbertoEdoardo I'm interested in a global proof without local methods. This is the classical approach. As for your last question, e.g. E.Artin's lecture on class field theoy(1932). The English translation by R.Friedman is in the appendix of H.Cohn's A classical invitation to algebraic numbers and class fields. – Makoto Kato Sep 14 '12 at 21:57
Hi Makoto: There are various ways that this question could be improved. I think a primary one is purely social: I bet many of us would be interested to hear a bit about why you are interested in this question. (I'm much more interested if I understand why you are!) Remember that MO is narrowly focused on research mathematics, and so the best answer to my "social" question would be that you are interested in a generalizing some result to a context in which certain techniques fail, etc. As it stands, it seems that you are simply asking to avoid a standard and useful tool for no good reason. – Theo Johnson-Freyd Sep 15 '12 at 4:59
@TheoJohnson-Freyd My question is motivated by the following question. I'm interested in proving the main theorems of global class field theory using only classical ideal theory. mathoverflow.net/questions/95225/… – Makoto Kato Sep 15 '12 at 6:35