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\def\CC{\mathbb{C}}
\def\FF{\mathbb{F}}
\def\PP{\mathbb{P}}
\def\QQ{\mathbb{Q}}
\def\RR{\mathbb{R}}
\def\ZZ{\mathbb{Z}}
\def\gotha{\mathfrak{a}}
\def\gothb{\mathfrak{b}}
\def\gothd{\mathfrak{D}}
\def\gothm{\mathfrak{m}}
\def\gotho{\mathfrak{o}}
\def\gothp{\mathfrak{p}}
\def\gothq{\mathfrak{q}}
\DeclareMathOperator{\Disc}{Disc}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\Norm}{Norm}
\DeclareMathOperator{\Trace}{Trace}
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\begin{document}

\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Lectures 32-33: Different and Discriminant \\
November 19-21, 2001
\end{center}

\head{Reference} Neukirch, Section III.2.

\head{Outline of lectures}

Unless otherwise specified, $L/K$ will be a finite extension either of number
fields or of finite extensions of $\QQ_p$.

\begin{enumerate}
\item
Define the \emph{relative norm} $\Norm_{L/K}(\gotha)$ of a
fractional ideal $\gotha$ of $L$ as the fractional ideal
of $K$ generated by 
$\Norm_{L/K}(\alpha)$ for all $\alpha \in \gotha$.
Notice that for $L/K$ an extension of number fields,
the relative norm can be computed locally:
\[
\Norm_{L/K}(\gotha) = \prod_\gothp \prod_{\gothq} \Norm_{L_\gothq/K_\gothp}
(\gotha)
= \prod_\gothp \prod_{\gothq} \gothp^{f(\gothq/\gothp) v_{\gothq}(\gotha)}.
\]
\item
For $L/K$ any finite extension of fields, the \emph{trace pairing}
$\langle \cdot, \cdot \rangle: L \times L \to K$ is given by
$\langle x,y \rangle = \Trace_{L/K}(xy)$.
The trace pairing is nondegenerate (i.e., $\langle x, y \rangle = 0$
for all $y$ implies $x=0$) if and only if $L/K$ is separable.
\item
Notice that
the trace pairing takes $\gotho_L \times \gotho_L$ into
$\gotho_K$. (The trace of an algebraic integer is the sum of
algebraic integers.)
\item
Define the \emph{inverse different} $\gothd^{-1}_{L/K}$ as the set
of $x \in L$ such that $\langle x,y \rangle \in \gotho_K$ for all
$y \in \gotho_L$. Note that this is a fractional ideal of $\gotho_L$.
\item
Define the \emph{different} $\gothd_{L/K}$
as the inverse of the inverse different; this is always an integral
ideal.
Note that $\gothd_{L/K} = \prod_{\gothp} \prod_{\gothq}
\gothd_{L_\gothq/K_\gothp}$. 
\item
If $M/L$ is another extension, note that
$\gothd_{M/K} = \gothd_{M/L} \gothd_{L/K}$, as an ideal of $\gotho_M$.
The point is that $\Trace_{L/K}(\Trace_{M/L}(x)) = \Trace_{M/K}(x)$,
so the trace pairing from $M$ to $K$ can be computed ``all at once''
or ``in two stages''.
\item
In case $\gotho_L = \gotho_K[\alpha]$ for some $\alpha \in \gotho_L$,
let $P(x)$ be the minimal polynomial of $\alpha$ over $\gotho_L$.
Show that $\gothd_{L/K} = (P'(\alpha))$ by showing that the dual basis
of $1, \alpha,\dots, \alpha^{n-1}$ is
\[
\frac{b_0}{P'(\alpha)}, \cdots, \frac{b_{n-1}}{P'(\alpha)},
\]
where $P(x)/(x-\alpha) = b_0 + b_1x + \cdots + b_{n-1}x^{n-1}$.
In the general case,
$\gothd_{L/K}$ is the ideal generated by $P'(\alpha)$ for all pairs
$\alpha, P$ of an element $\alpha$ of $\gotho_L$ which generates $L$ over
$K$, and the minimal polynomial $P$ of $\alpha$.
\item
Show that the relative
discriminant is the relative norm of the different, by noting that
$\Disc(1, \alpha, \dots, \alpha^{n-1}) =
\Norm_{L/K} (P'(\alpha))$. (For another approach, see Neukirch,
Theorem III.2.9.)
\item
Let $\gothq$ be a prime of $L$ over $\gothp$ and let $e = e(\gothq/\gothp)$
be the ramification index. Show that if $\gothq/\gothp$
is unramified or tamely ramified, then $v_{\gothq}(\gothd_{L/K})
= e-1$; otherwise, $e \leq v_{\gothq}(\gothd_{L/K}) \leq e-1+v_{\gothq}(e)$.
In particular, $\gothq$ is ramified over $\gothp$ if and only if
$\gothq$ divides $\gothd_{L/K}$.
\end{enumerate}

\end{document}