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\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Lecture 24: Ramification in Galois Extensions \\
October 29, 2001
\end{center}

\head{Reference}
Neukirch, Section 1.9; optionally, Section 2.9 as well.

\head{Jargon watch}
Given an extension $L/K$ of fields, the \emph{normal closure} of $L/K$
is the compositum of the image of $L'$ under all elements of
$\Gal(\overline{K}/K)$. If this equals $L$, we say $L/K$ is \emph{normal}.
We say $L/K$ is \emph{Galois} if it is finite, normal and separable,
or equivalently, if $\#\Gal(L/K) = [L:K]$.
Note that finite and separable are automatic for $L/K$ an extension either
of number fields or of finite extensions of $\QQ_p$.

\head{Outline of lecture}

\begin{enumerate}
\item
Let $L/K$ be a Galois extension of number fields with Galois group $G$.
Show that $G$ acts transitively on the primes $\gothb$
of $\gotho_L$ lying above a prime $\gothp$ of $\gotho_K$. Note in particular
that they all have the \emph{same} ramification index and inertia degree
over $\gothp$.
\item
Let $\gothb$ be a prime of $\gotho_L$ lying over the prime $\gothp$ of
$\gotho_K$.
Define the \emph{decomposition group} $G_{\gothb}$ of $\gothb$ over $K$
as the set of $\sigma \in G$ mapping $\gothb$ to
itself (but not necessarily by the identity map!). Define the fixed field
of $G_{\gothb}$ as the \emph{decomposition field} of $\gothb$ over $K$.
\item
Interpret some of the splitting behavior of $\gothp$ in terms of the
decomposition group. E.g., $G_{\gothb} = 1$ if and only if $\gothp$ is
totally split.
\item
Show that $G_{\gothb}$ surjects onto the Galois group 
$\Gal((\gotho_L/\gothb)/(\gotho_K/\gothp)$. The kernel of this map
is the \emph{inertia group} $I_{\gothb}$ of $\gothb$ over $K$,
whose fixed field is the \emph{inertia field} of $\gothb$ over $K$.
Again, interpret splitting behavior in terms of this: the order of
$I_{\gothb}$ is the ramification index $e(\gothb/\gothp)$, and the
order of $G_{\gothb}/I_{\gothb}$ is the inertia degree.
\item
Note that $G_{\gothb}$ can also be interpreted as the Galois group
of the extension $L_\gothb/K_\gothp$ of local fields, and that its
subgroup $I_{\gothb}$ is the Galois group of $L_{\gothb}$ over
its maximal unramified subextension (over $K_{\gothp}$). (Of course,
one can also define the inertia subgroup for any Galois extension of
local fields.)
\item
Point out that finite Galois extensions of local fields are always
solvable, so that decomposition groups are always solvable.
More precisely, the decomposition group is the semidirect product
of the inertia group by the Galois group of the residue field extension,
the inertia group is the semidirect product of the ``wild inertia group''
by a cyclic group, and the wild inertia group is a $p$-group, where $p$
is the characteristic of the residue field.
\item
If time permits, mention how this all can be interpreted for a non-Galois
extension. Namely, if $L/K$ is not Galois, let $N/K$ be its normal closure,
put $G = \Gal(N/K)$, and put $H = \Gal(N/L)$. Now for $\gothb$ a prime
of $N$ above $\gothp$, the double cosets $H \backslash G / G_{\gothb}$
correspond to the primes of $L$ above $\gothb$, and one can relate
the splitting of $\gothp$ to this decomposition: for examples, see the
homework.
\end{enumerate}

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