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\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Lecture 20: Ramification in Number Fields\\
October 12, 2001
\end{center}

\head{Reference}
Neukirch, Section 2.8, and to a lesser extent Section 1.8. 

\head{Outline of lectures}

\begin{enumerate}
\item
(Left over from last time)
Work out the structure of $\QQ_p(\zeta_{p^n})$. In particular, show that
$1-\zeta_{p^n}$ generates the valuation ring.
\item
Let $L/K$ be an extension of \emph{number fields}, let $\gothq$ be a
nonzero prime ideal of $\gotho_L$, and let $\gothp = K \cap \gothq$.
As noted earlier, the completion $L_\gothq$ is a finite extension of
$K_\gothp$. Define the ramification index and inertia degree
$e(\gothq/\gothp) = e(L_{\gothq}/K_{\gothp})$
and $f(\gothq/\gothp) = f(L_{\gothq}/K_{\gothp})$.
\item
Prove that the $\gothp$-adic valuation does not necessarily extend uniquely
to $L$, in contrast to the case where $K$ was a local field. Rather,
the extensions are precisely the $\gothq$-adic valuations for all
primes $\gothq$ containing
$\gotho_L \gothp$, or
equivalently, all primes of $\gotho_L$ in the prime factorization
of $\gotho_L \gothp$. (The point: given an extension of the valuation
to $L$, the maximal ideal of the valuation ring is a prime ideal
containing $\gotho_L \gothp$.)
\item
For $L/K$ a finite extension of number fields and $\gothp$ a prime
of $\gotho_K$, prove the fundamental identity
\[
[L:K] = \sum_{\gothq} [L_{\gothq}:K_{\gothp}] = 
\sum_{\gothq} e(\gothq/\gothp) f(\gothq/\gothp),
\]
where the sum runs over all primes $\gothq$ of $\gotho_L$ 
containing $\gothp$.
\item
Point out another interpretation: the tensor product $L \otimes_K K_{\gothp}$
factors as the product of the $L_{\gothq}$. 
\item
Prove that $\gotho_L\gothp = \prod_{\gothq}
\gothq^{e(\gothq/\gothp)}$. That is, the 
ramification index is the factor to which $\gothq$ shows up in the prime
factorization of $\gotho_L \gothp$.
\item
Discuss some examples, e.g., $\QQ(i)$ and/or
$\QQ(x)/(x^3+x+1)$. Point out (to be proved later)
that when $K=\QQ$, the only primes of $K$ over which ramification happens
are the ones dividing the discriminant of $L$. (One part of this claim is
on this week's homework.) Later we will define the ``relative discriminant'',
which will allow to make a similar statement for general $K$.
\end{enumerate}

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