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\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Lectures 18-19: Ramification \\
October 8 and 10, 2001
\end{center}

\head{Reference}
Neukirch, Sections 2.6 and 2.7.

\head{Jargon watch}
Reminder: a \emph{local field} is a field which is complete with respect
to a discrete nonarchimedean valuation. You may substitute ``finite
extension of $\QQ_p$'' if you like.

\head{Outline of lectures}

\begin{enumerate}
\item
(Left over from last time)
Define the Newton polygon of a polynomial over a local field,
and show that it computes the valuations of the roots of the polynomial.
Note that an irreducible polynomial has one slope (by the extension
theorem, all roots of the polynomial must have the same valuation).
\item
Given a finite extension $L/K$ of local fields, define the \emph{ramification
index} $e = e(L/K)$ as the index of the value group of $K$ in the value group
of $L$. That is, if $\pi$ generates the maximal ideal of the valuation ring
of $L$, then $\pi^e$ generates the maximal ideal of the valuation ring of $K$.
\item
Given a finite extension $L/K$ of local fields, define the \emph{inertia
degree} $f = f(L/K)$ as the degree of the residue field of $L$ over the
residue field of $K$.
\item
Prove that if $K$ is a local field, then
$e(L/K) f(L/K) = [L:K]$ (the ``fundamental identity'').
Illustrate this with $\QQ_p(i)$ and possibly other examples.
\item
Define an \emph{unramified} extension $L/K$ of local fields
as one in which $e(L/K) = 1$; that is, the residue field extension
has degree $[L:K]$. Note that if $K'/K$ is any finite extension,
then the compositum $K'L$ is unramified over $K'$. Also note that
a local field has only one unramified extension of any given degree
(see homework).
At the other extreme, define a \emph{totally ramified} extension
to be one where $f(L/K) = 1$.
\item
Distinguish between tamely ramified and wildly ramified extensions.
An extension $L/K$ of local fields, whose residue fields have characteristic
$p$, is \emph{tamely ramified} if its ramification degree is not divisible
by $p$. Show that tamely ramified extensions are generated by radicals.
An example of a wildly ramified extension: $\QQ_p(p^{1/p})$.
\item
Illustrate with the example $\QQ_p(\zeta_n)$, where $\zeta_n$ is an $n$-th
root of unity.
\end{enumerate}

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