Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Here is a problem in charpter $2$ section $5$ in Algebraic Number Theory written by Jiirgen Neukirch.

The problem is

Let $K$ be a $p-adic$ number field, $v_p$ the normalized exponential valuation of $K$, and $dx$ the Haar measure on the locally compact additive group $K$, scaled so that $\int_O dx = 1$. Then one has $v_p(a) = \int_{aO} dx$.

I would like to ask why the last equation holds,I think it should be $\shortmid a\shortmid _p =\int_{aO} dx$.

Please help me with it,thank you very much.

share|improve this question
1  
Dear curiosity, For those without Neukirch's book on hand, could you briefly explain what is the normalized exponential valuation. (E.g. is it an absolute value, taking products to products, or a valuation in the ring-theoretic sense, taking products to sums?) In other words, what is the difference between $v_p(a)$ and $|a|_p$? Regards, –  Matt E May 26 '11 at 18:29
    
I am sorry for making you confused, here $v_p$ means taking products to products while the other means the usual valuation in the ring-theoretic sense. –  curiosity May 27 '11 at 4:41

1 Answer 1

up vote 3 down vote accepted

The question is equivalent to asking for the ratio of the measures $\mu(a \mathcal{O})/\mu(\mathcal{O})$, where $\mathcal{O}$ is the valuation ring and $a \in K^{\times}$. But in a $p$-adic field, whenever you have one ball centered at zero contained in another ball centered at $0$, the larger ball is simply a disjoint union of translates of the smaller ball. Since the Haar measure is translation invariant, the ratio of the volumes just comes out to be the number of translates. If $a \in \mathcal{O}$, then the number of translates of $a \mathcal{O}$ in $\mathcal{O}$ is nothing else than the cardinality of the quotient ring $\mathcal{O}/a \mathcal{O}$.

I leave it to you to work out the problem according to the notation in the book (especially since I think your comment trying to clarify the notation may be mistaken). But for instance in $\mathbb{Q}_p$, the ratio of the volume of $\mathcal{O}$ to $p \mathcal{O}$ is equal to $p$, because $\# \mathcal{O}/p\mathcal{O} = \# \mathbb{Z} / p \mathbb{Z} = p$. In general this ratio will be multiplicative in $a$, not additive (so it should be a $K$-adic norm, not a $K$-adic valuation).

share|improve this answer
    
@Peter: Thank you,I will take a try. –  curiosity May 27 '11 at 12:19
    
@Peter:I hane worked it out,thank you very much. –  curiosity May 30 '11 at 17:41

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.