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I'm trying to figure out how to compute higher ramification groups of $\mathbb{Q}_p(\zeta_{p^a})$. I've tried to proceed as follows. Write $L= \mathbb{Q}_p(\zeta_{p^a})$ and $K=\mathbb{Q}_p$, so we know that $G=\textrm{Gal}(L/K)\simeq (\mathbb{Z}/p^a\mathbb{Z})^\times$. Since the extension is totally ramified we immediately have

$$G_{-1}=G,\;\; G_0=G$$

and we also know that $G_1$ is the $p$-Sylow subgroup of $G$. To compute $G_i$ for $i\geq 1$ we know that we have an injective homomorphism given by

$$G_i/G_{i+1}\hookrightarrow U_L^{(i)}/U_L^{(i+1)}\simeq \overline{L},\;\;\sigma \mapsto \frac{\sigma \pi_L}{\pi_L}$$

where the uniformizer of $L$ is $\pi_L=1-\zeta_{p^a}$. Writing $\sigma_t$ for the element of $G$ s.t. $\zeta_{p^a}\mapsto \zeta_{p^a}^t$ we have that $\sigma_t\in G_i$ satisfies

$$\sigma_t\in G_{i+1}\Leftrightarrow \frac{\sigma_t \pi_L}{\pi_L}=\frac{1-\zeta_{p^a}^t}{1-\zeta_{p^a}}=1+\zeta_{p^a}+\ldots+\zeta_{p^a}^{t-1}\equiv 1\,(\textrm{mod }\pi_L^{i+1})$$

I'm stuck here for the case when $i>0$, since $\pi_L^{i+1}$ seems like an annoying element to work with and I can't seem to figure out how to represent $\zeta_{p^a}$ modulo $\pi_L^{i+1}$.

Any ideas on how to proceed? I'm not necessarily looking for a complete solution more like hints. I'm also curious if there are any other approaches to this problem that might work as I'm interested in general tricks that apply to computing higher ramification groups.

EDIT: I looked at Neukirch and he assigns the same problem at the end of chapter 2. Up to that point the book has covered nothing except classical theory of number fields and basic theory of valued fields. I'm more or less looking for hints in this context, though I can come back to more advanced hints once I have time to study that theory.

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Here's a (possibly rather obscure) hint: the Artin conductor of a character of G is going to be exactly what you think it is! –  David Loeffler Jan 2 '12 at 11:44
    
@DavidLoeffler: Yes, the text that I'm studying doesn't even define the Artin conductor, so that's definitely a bit obscure. :) –  pki Jan 2 '12 at 11:53
    
See en.wikipedia.org/wiki/Artin_conductor. From my (rather biased) viewpoint I'd say that defining the Artin conductor is one of the most important applications of higher ramification theory, so it's bizarre that your text doesn't mention it. –  David Loeffler Jan 2 '12 at 11:54
    
It's not a textbook it's a lecture note on the theory of local fields. –  pki Jan 2 '12 at 11:55
    
I tried to keep my hint relatively minimal. Let me know if you'd like more details. –  Cam McLeman Jan 3 '12 at 16:35

1 Answer 1

up vote 2 down vote accepted

A hint: Let $v_L$ denote the valuation in $L$. Then just by clearing denominators in your formula, you have $$ \sigma_t\in G_i \iff v_L(\sigma_t(\pi_L)-\pi_L)\geq i+1. $$ Now given your explicit form of $\pi_L$ and $\sigma_t$ (long live cyclotomic extensions!), it's not hard to determine this valuation -- it's now reduced to determining $i$ in terms of orders of $t$ modulo various powers of $p$ less than $p^a$, etc.

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