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\begin{proof}
  By \cref{prop_13_10}, \cref{thm_14_2} and
  \cref{thm_13_4}, for $s \in \Zbb_{\ge -1}$,
  \[
    G_s / G_{s + 1} \cong \text{a subgroup} \begin{cases}
      \Gal(k_L / k) & \text{if $s = -1$} \\
      (k_L^\times, \times) & \text{if $s = 0$} \\
      (k_L, +) & \text{if $s \ge 1$}
    \end{cases}
  \]
  Thus $G_s / G_{s + 1}$ is solvable for $s \ge
  -1$. Conclude using \cref{thm_14_2}(ii).
\end{proof}

Let $\char k = p$. Then $p \nmid |G_0 / G_1|$ and
$|G_1| = p^n$. Thus $G_1$ is the unique (since
normal) Sylow $p$-subgroup of $G_0 = \I_{L / K}$.

\begin{fcdefn}[]
  \glsnoundefn{wig}{wild inertial group}{wild
  inertial groups}%
  \glsnoundefn{tquot}{tame quotient}{tame
  quotients}%
  \label{defn_14_4}
  % Definition 14.4
  $G_1$ is called the wild inertial group, and
  $G_0 / G_1$ is called the tame quotient.
\end{fcdefn}

\glsadjdefn{tram}{tamely ramified}{extension}%
\glsadjdefn{wram}{wildly ramified}{extension}%
Suppose $L / K$ is finite separable. Say $L / K$
is tamely ramified if $\char k \nmid e_{L / K}$.
Otherwise it is wildly ramified.

\begin{fcthm}[]
  \label{thm_14_5}
  % Theorem 14.5
  Assuming:
  - $[K : \Qp] < \infty$
  - $L / K$ finite
  - $\diff LK = (\pi^{\delta(L / K)})$
  Then:
  $\delta(L / K) \ge e_{L / K} - 1$, with equality
  if and only if \gls{tram}. In particular, $L /
  K$ \gls{unramext} if and only if $\diff LK
  = \O_L$.
\end{fcthm}

\begin{proof}
  Example Sheet 3 shows $\diff LK = \diff L{K_0}
  \cdot \diff{K_0}K$. Suffices to check $2$ cases:
  \begin{enumerate}[(i)]
    \item $L / K$ \gls{unramext}. Then
      \cref{prop_6_12} gives that $\O_L =
      \O_K[\alpha]$, for some $\alpha \in \O_L$
      with $k_L = k(\ol{a})$.

      Let $g(X) \in O_K[X]$ be the minimal
      polynomial of $\alpha$. Since $[L : K] =
      [k_L : k]$, we have that $\ol{g}(X) \in
      k[X]$ is the minimal polynomial of $\ol{a}$.
      $\ol{g}(X)$ separable and hence
      $g'(\alpha) \not\equiv 0 \pmod \pi_L$.
      \cref{thm_12_8} implies $\diff LK =
      (g(\alpha)) = \O_L$.
    \item $L / K$ \gls{totramext}. Say $[L : K] =
      e$, $\O_L = \O_K[\pi_L]$, $\pi_L$ a root of
      \[
        g(X) = X^e + \sum_{i = 0}^{e - 1} a_i X^i
        \in \O_K[X]
      \]
      is \gls{eispoly}. Then
      \[
        g'(\pi_L)
        = \ub{e\pi_L^{e - 1}}_{\ge e - 1} +
        \ub{\sum_{i = 1}^{e - 1} i a_i \pi_L^{i -
        1}}_{v_L \ge e}
      .\]
      Thus $v_L(y'(\pi_L)) \ge e - 1$. Equality
      if and only if $p \nmid e$. \qedhere
  \end{enumerate}
\end{proof}

\begin{corollary}
  \label{coro_14_6}
  % Corollary 14.6
  Suppose $L / K$ is an extension of number
  fields. Let $P \subseteq \O_L$, $P \cap \O_K =
  \pideal$. Then $e(P / \pideal) > 1$ if and only
  if $P \mid \diff LK$.
\end{corollary}

\begin{proof}
  \cref{thm_12_9} implies $\diff LK = \prod_P
  \diff{L_P}{K_\pideal}$. Then use $e(P / \pideal)
  = e_{L_P / K_\pideal}$ and \cref{thm_14_5}.
\end{proof}

\begin{example*}
  \begin{itemize}
    \item $K = \Qp$, $\zeta_{p^n}$ a primitive $p^n$-th
      root of unity. $L = \Qp(\zeta_{p^n})$. The
      $p^n$-th cyclotomic polynomial is
      \[
        \Phi_{p^n}(X) = X^{p^{n - 1}(p - 1)}
        + X^{p^{n - 1}(p - 2)} + \cdots + 1 \in \Zp[X]
      .\]
      See Example Sheet 3.
    \item $\Phi_{p^n}(X)$ irreducible (hence
      $\Phi_{p^n}(X)$ is the minimal polynomial of
      $\zeta_{p^n}$).
    \item $L / \Qp$ is Galois, \gls{totramext} of
      degree $p^{n + 1}(p - 1)$.
    \item $\pi \defeq \zeta_{p^n} - 1$ a
      \gls{unif} in $\O_L$ $\leadsto$ $\O_L =
      \Zp[\zeta_{p^n} - 1] = \Zp[\zeta_{p^n}]$.
    \item $\Gal(L / \Qp) \stackrel{\sim}{\to}
      (\Zbb / p^n \Zbb)^\times$ (abelian).
      $\sigma_m \leftrightarrow m$ where
      $\sigma_m(\zeta_{p^n}) = \zeta_{p^n}^m$.
      \[
        v_L(\sigma_m(\pi) - \pi)
        = v_L(\zeta_{p^n}^m - \zeta_{p^n})
        = v_L(\zeta_{p^n}^{m - 1} - 1)
      .\]
      Let $k$ be maximal such that $p^k \mid m -
      1$. Then $\zeta_{p^n}^{m - 1}$ is a
      primitive $p^{n - k}$-th root of unity, and
      hence $\zeta_{p^n}^{m - 1} - 1$ is a
      \gls{unif} $\pi'$ in $L' =
      \Qp(\zeta_{p^n}^{m - 1})$. Hence
      \[
        v_L(\zeta_{p^n}^{m - 1} - 1)
        = e_{L / L'}
        = \frac{e_{L / \Qp}}{e_{L' / \Qp}}
        = \frac{[L : \Qp]}{[L' : \Qp]}
        = \frac{p^{n - 1}(p - 1)}{p^{n - k - 1}(p
        - 1)}
        = p^k
      .\]
      \cref{thm_14_2}(i) implies that $\sigma_m
      \in G_i$ if and only if $p^k \ge i + 1$.
      Thus
      \[
        G_i \cong \begin{cases}
          (\Zbb / p^n\Zbb)^\times & i \le 0 \\
          (1 + p^k \Zbb) / p^n \Zbb & p^{k - 1} -
          1 < i \le p^k - 1 (1 \le k \le i + 1) \\
          \{1\} & p^{n - 1} - 1 < i
        \end{cases}
      .\]
  \end{itemize}
\end{example*}

\newpage
\part{Local Class Field Theory}

\newpage
\section{Infinite Galois Theory}

\begin{fcdefn}[Infinite Galois definitions]
  \label{defn_16_1}
  % Definition 16.1
  \begin{itemize}
    \item $L / K$ is separable if $\forall \alpha
      \in L$, the minimal polynomial $f_\alpha(X)
      \in K[X]$ for $\alpha$ is separable.
    \item $L / K$ is normal if $f_\alpha(X)$
      splits in $L$ for all $\alpha \in L$.
    \item $L / K$ is Galois if it is separable and
      normal.
      Write $\Gal(L / K) \defeq \Aut_K(L)$ in this
      case. If $L / K$ is a finite Galois
      extension, then we have a Galois
      correspondence:
      \begin{align*}
        \{\text{subextensions $K \subseteq K'
        \subseteq L$}\}
        &\leftrightarrow \{\text{subgroups of
        $\Gal(L / K)$}\} \\
        K'
        &\mapsto \Gal(K / K')
      \end{align*}
  \end{itemize}
\end{fcdefn}

Let $(I, \le)$ be a poset. Say $I$ is a directed
set if for all $i, j \in I$, there exists $k \in
I$ such that $i \le k$, $j \le k$.

\begin{example*}
  \phantom{}
  \begin{itemize}
    \item Any total order (for example $(\Nbb,
      \le)$).
    \item $\Nbb_{\ge 1}$ ordered by divisibility.
  \end{itemize}
\end{example*}

\begin{fcdefn}
  \label{defn_16_2}
  % Definition 16.2
  Let $(I, \le)$ be a directed set and $(G_i)_{i
  \in I}$ a collection of groups together with
  maps $\varphi_{ij} : G_j \to G_i$, $i \le j$
  such that:
  \begin{itemize}
    \item $\varphi_{ik} = \varphi_{ij} \circ
      \varphi_{jk}$ for any $i \le j \le k$
    \item $\varphi_{ii} = \id$
  \end{itemize}
  Say $((G_i)_{i = 1}, \varphi_{ij})$ is an
  inverse system. The inverse limit of $(G_i,
  \varphi_i)$ is
  \[
    \displaystyle \lim_{\stackrel[i]{}{\longleftarrow}}
    G_i = \{(g_i)_{i \in I} \in \prod_{i \in I}
    G_i \mid \varphi_{ij}(g_j) = g_i\}
  .\]
\end{fcdefn}

\begin{remark*}
  \phantom{}
  \begin{itemize}
    \item $(\Nbb, \le)$ recovers the previous set.
    \item There exist projection maps $\varphi_j :
      \dinvlim{i \in I} G_i \to G_j$.
    \item $\dinvlim{i \in I} G_i$ satisfies a
      universal property.
    \item Assume $G_i$ finite. Then the profinite
      topology on $\dinvlim{i \in I} G_i$ is the
      weakest topology such that $\varphi_j$ are
      continuous for all $j \in I$.
  \end{itemize}
\end{remark*}

\begin{fcprop}[]
  \label{prop_16_3}
  % Proposition 16.3
  Assuming:
  - $L / K$ Galois
  Then:
  \begin{enumerate}[(i)]
    \item The set $I = \{F / K \text{finite} \st F
      \subseteq L,  \text{ $F$ Galois}\}$ is a
      directed set under $\subseteq$.
    \item For $F, F' \in I$, $F \subseteq F'$
      there is a restriction map $\res_{F, F'} :
      \Gal(F' / K) \surjto \Gal(F / K)$ and the
      natural map
      \[
        \Gal(L / K) \to \dinvlim{F \in I} \Gal(F
        / K)
      \]
      is an isomorphism.
  \end{enumerate}
\end{fcprop}

\begin{proof}
  Example Sheet 4.
\end{proof}