Higher Ramification Groups of Degree $p$ Cyclic Extensions.

Question

For the person who is about to report this as a duplicate, "no, this isn't a duplicate".

So I am trying to prove the following two (related) problems:

Suppose $K$ contains a primitive $p$th root of unity. Let $L=K[x]/(x^p-\pi)$ where $\pi$ is the prime in $\mathcal{O}_K$. Then $L$ is a cyclic totally ramified extension of $K$.

Show that if $s\in G(L/K)=:G$ such that $\langle s\rangle=G$, then $s\in G_i$ and $s\notin G_{i+1}$ where $i=e_L/(p-1)$ where $e_L$ is the absolute ramification index of $L$. (In other words, show $i=e_L/(p-1)$ is the first jump index)

Let $e_K$ be the absolute ramification index of $K$, and let $n$ be a positive integer prime to $p$ and $n< pe_K/(p-1)$. Let $y$ be an element of valuation $-n$. Define $$ S(x)=x^p-x-y $$ and let $L=K[x]/(S(x))$. Show that $G_n=G(L/K)$ and $G_{n+1}=\{1\}$.

Essentially, my end goal here is understanding the following paper: http://www.jstor.org/stable/2372667 (for those interested)

Answer 1

$x^p-\pi$ is Eisenstein and irreducible, thus $L/K$ is a degree $p$ extension, i.e. cyclic. It's totally ramified since $x^p-\pi$ is Eisenstein. By exer 2.3.c in Serre, $G_{i+1}=\{1\}$. Consider the chain of extensions $\mathbb{Q}_p\subseteq \mathbb{Q}_p(\zeta)\subseteq K\subseteq L$. By definition, $$e=v_L(p)=[L:K]e_{K/\mathbb{Q}_p[\zeta]}[\mathbb{Q}_p(\zeta):\mathbb{Q}_p]v_{\mathbb{Q}_p}(p)=p(p-1)e_{K/\mathbb{Q}_p[\zeta]},$$ so $$ v_L(\zeta\pi^{1/p}-\pi^{1/p})=1+v_L(\zeta-1)=1+pe_{K/\mathbb{Q}_p[\zeta]}=1 + i $$ so $G_{i}$ is non-trivial, i.e. $G_i=G(L/K)$.

Answer 2

If you agree with the setting: $K$ is a "local field with unequal characteristic", i.e. a complete field w.r.t. a discrete valuation, with characteristic 0 and residual characteristic $p$, then your two questions can be found as exercises (with indications) in Serre's "Local Fields", chap. IV, §2, ex. 4 and 5. Ex. 4 refers to preliminary results of ex. 3, but a direct proof is as follows in case $K$ is a finite extension of $\mathbf Q_p$: let $e=e_K$ be the absolute ramification index of $K$; since $K$ contains the group $\mu_p$ of $p$-th roots of unity, $e$ is a multiple of $p-1= e(\mathbf Q_p (\mu_p)/\mathbf Q_p)$, so $e':=e/p-1$ is an integer. Let $\pi$ be a uniformizer of $K$. The Kummer extension $L=K(\sqrt [p] \pi)$ is cyclic of degree $p$, totally ramified, so its Galois group $G$ admits a single ramification "break number" $b$, i.e. the descending chain of ramification subgroups is $G=G_0 = ...=G_b > G_{b+1}=(1)$. If $g$ is a generator of $G$, by definition $b=ord_L(g(\sqrt [p] \pi)/\sqrt [p] \pi -1)= ord_L (\zeta - 1)=pe'$ where $\zeta$ is a primitive $p$-th root of unity . The last equality holds because ($\zeta - 1$) is a uniformizer of $\mathbf Q_p (\mu_p)$.

NB. The above result can be generalized to compute the break number of a Kummer extension $L=K(\sqrt [p] a)$, with $a\in K^* - (K^*)^{p}$. Define the "defect" $def_K (\bar a)$, where $\bar a$ is the class of $a$ mod $(K^*)^{p}$, to be $sup. ord_K (1-x)$ for all $x \in \bar a$ . A result of B. Wyman, "Wildly ramified Gamma extensions", Amer. J. of Math., 91,1 (1969), 135-152 states that the values of the function $def_K$ are all the integers $j$ s.t. $1\le j \le pe'$ and $(p,j)=1$ if the right inequality is strict; moreover, $def(\bar a)=pe'$ iff $L/K$ is unramified, and in the other cases, $L/K$ is totally ramified, with break number $b= pe'- def(\bar a)$ .