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\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Lectures 21--22: Ramification and Splitting in Number Fields\\
October 22 and 24, 2001
\end{center}

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

\head{Notation}
For $L/K$ an extension of number fields, $\gothp$ a prime of
$\gotho_K$ and $\gothq$ a prime of $\gotho_L$ containing
$\gothp \gotho_L$, we defined
last time the ramification index and inertia degree
$e(\gothq/\gothp) = e(L_{\gothq}/K_{\gothp})$
and $f(\gothq/\gothp) = f(L_{\gothq}/K_{\gothp})$.

\head{Outline of lectures}

\begin{enumerate}
\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 $\gothp$. Proof outline: write $L = K(\alpha)$ by the primitive
element theorem, where $\alpha$ has minimal polynomial $P(x)$, and
look at the factorization of $P(x)$ over $K_\gothp$. Each factor corresponds
to a prime $\gothq$ 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
$\QQ(x)/(x^3+x+1)$.
\item
As an additional example,
mention the law of quadratic reciprocity (Theorem 8.6). To be done a bit later,
maybe Friday:
discuss the proof given in Neukirch section I.8 (using Gauss sums).
\item
If $L = K(\alpha)$ for some $\alpha \in \gotho_L$, define
the \emph{conductor} of $\alpha$ as the ideal of $\beta \in \gotho_K$
such that $\beta \gotho_K \subseteq \gotho_K[\alpha]$. Note that
if $\gothp$ is coprime to the conductor of $\alpha$, the ramification
and inertia degrees are computed by the prime factorization of the minimal
polynomial of $\alpha$ modulo $\gothp$. (Namely, each factor of degree $d$
corresponds to a prime above $\gothp$ with inertia degree $d$, and its
multiplicity is the ramification degree of the prime.)
\item
Important special case: if $\gotho_L = \gotho_K[\alpha]$, then the conductor
is the unit ideal in $\gotho_K$, so the above criterion applies for all
primes.
This is useful for determining splitting, but pretty useless for
determining ramification, since you need to compute discriminants to
find the ring of integers in $\gotho_L$; instead, you should factor the minimal
polynomial of $\alpha$ over the completion $K_{\gothp}$.
\item
A more useful form of the splitting criterion: if $L/K$ is an extension
of number fields, $\alpha \in \gotho_L$ generates $L$ over $K$ and
$\gothp$ is coprime to the discriminant of the minimal polynomial of
$\alpha$ over $K$, then the degrees of the factors of the minimal polynomial
modulo $\gothp$ are the inertia degrees of the primes above $\gothp$ in $L$,
all of which are unramified.
\item
If $L/K$ is an extension of local fields, define the
\emph{relative discriminant} $\disc(L/K)$ as the 
ideal of $\gotho_K$ generated by the determinant of the matrix
$\Trace(x_i x_j)_{i,j=1}^n$, for $x_1, \dots, x_n$ a basis of
$\gotho_L$ over $\gotho_K$. Caution: the relative discriminant is only
defined as an ideal, not as an element.
\item
If $L/K$ is an extension of number field, define the relative
discriminant 
\[
\disc(L/K) = \prod_{\gothp} \prod_{\gothq} \disc(L_\gothq/K_\gothp).
\]
Here $\gothp$ runs over all nonzero primes of $\gotho_K$,
$\gothq$ runs over all primes of $\gotho_L$ containing $\gothp
\gotho_L$, and we interpret $\disc(L_\gothq/K_\gothp)$ to be a power
of $\gothp$ even though it's really a power of the maximal ideal
of $K_\gothp$.
\item
Note that for $K=\QQ$, the relative discriminant $\disc(L/K)$ is the ideal
generated by the usual discriminant of $L$.
\end{enumerate}

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