Timeline for Relation between ramification index and length of filtration of ramification groups

5 events

when toggle format	what		by	license	comment
Mar 27, 2016 at 7:23	comment	added	MiRi_NaE		Dear nfdc23, I appreciate for the case being $G_n$ nontrivial for sufficiently large $n$. Your example in Serre's 'Local Fields' requires large absolute ramification index $e(K/\mathbb{Q}_p )$. It's interesting. But I also want to find example for a fixed $K$, in special when $K$ absolutely unramified. Fortunately, I've found remarks after proposition13 of section 6 of Chapter 3 of Serre's 'local fields', saying upper and lower bound of different. This gives asymptotic answer for the case I've mentioned. Thanks again!
Mar 24, 2016 at 4:59	history	edited	MiRi_NaE	CC BY-SA 3.0	edited title
Mar 24, 2016 at 2:28	comment	added	nfdc23		There is no bound in terms of $[L:K]$ when working with $p$-adic fields for a fixed prime $p$. Exercise 5 of section 2 of Chapter IV of Serre's book "Local Fields" (the ur-reference on all matters related to the basic ramification theory of local fields) shows that for fixed $p$ and any $n > 0$ if we consider extensions $K$ of $\mathbf{Q}_p$ for which $e(K/\mathbf{Q}_p)$ is large enough (in terms of $n$ and $p$) then $K$ admits a degree-$p$ cyclic extension for which $G_n=G$ and $G_{n+1}=1$. I strongly recommend working through all of the Exercises in that section; very instructive!
Mar 24, 2016 at 2:20	review	First posts
Mar 24, 2016 at 2:29
Mar 24, 2016 at 2:15	history	asked	MiRi_NaE	CC BY-SA 3.0