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I have been doing some revision on local field theory and have gathered up a collection of questions which I have been unable to make much progress with; there will be a few similar queries along with this problem, I hope that's ok. I wish to prove the following:

Suppose $K$ is a field that is complete with respect to a discrete valuation $v$, and $K$ has finite residue field $k$, say. Prove that when $\operatorname{char}k \nmid n$, it holds that $|K^*/(K^*)^n| = n|\mu_n(K)|$, where $\mu_n(K)$ is the group of $n$-th roots of unity in $K$.

I think this is one of the easier problems I am struggling with. I think notation and terminology is largely standard here, ask if not.

Now intuitively I can sort of see how this all ties together; working in $K^*$ modulo $(K^*)^n$ gives us a collection of cosets $a_i(K^*)^n$ and of course every one of these cosets will have order $m \leq n$ (and indeed $m \mid n$); so it's possible the right hand side of the equality is simply the product of the size of a coset (i.e. the size of $(K^*)^n$ ) and the number of such cosets.

I'm sure $\mu_n$ will come in in some obvious sense because generally if some property is true for $a$ (with regards to its $n$-th power) then it will probably also be true for $\zeta_n a$ whenever $\zeta_n \in \mu_n$. However, I'm afraid I've been unable to tie it all together. It is easy to see that the valuation of any element of $(K^*)^n$ must be divisible by $n$, but we haven't really used the residue field at all yet. If someone would be so kind as to help explain the steps I should be attempting to prove this (and ideally also explain why it makes sense to do this, if that isn't clear) then I'll be happy to give it a go myself.

You can assume I am probably familiar with most basic results on valuations, residue fields etc, I just don't know how to approach the problem. Many thanks as always.

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Instead of "revision" you probably mean "review", and the phrase "local field theory" sounds weird in English. ("Field theory" and "local class field theory" are standard, but "local field theory" is not, just like "complex analysis theory" is not used.) I think it is better to say you are reviewing local fields. –  KCd Jul 23 '12 at 10:06
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1 Answer

The major steps for a proof are the following:

  1. $K$ is the fraction field of a discrete valuation ring $O$ with prime element $t$, say, therefore every $x\in K^\ast$ can be written as $ut^m$, $m\in\mathbb{Z}$, $u\in O^\ast$. This gives information about the form of the elements of $K^\ast /(K^\ast)^n$.

  2. Since $O$ is complete one can use Hensel's lemma to show that units $u\in O^\ast$ such that the residue is an $n$-th power, are $n$-th powers themselves.

  3. Use the fact that the residue field $k$ is finite to compute the order of $k^\ast /(k^\ast)^n$.

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