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\def\CC{\mathbb{C}}
\def\FF{\mathbb{F}}
\def\PP{\mathbb{P}}
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\begin{document}

\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Lectures 9-10: The Class Number \\
September 17 and 19, 2001
\end{center}

\head{Reference}
Neukirch, Sections 1.5 and 1.6.

\head{Jargon watch}
The \emph{absolute norm} norm of an integral ideal $\gotha$ is equal to
the index $[\gotho_K:\gotha]$, i.e., the number of elements of the quotient
ring $\gotho_K/\gotha$. (Warning: the book uses parentheses instead
of brackets to denote the index.) Note that the absolute norm of a principal
ideal $(\alpha)$ is equal to the absolute value of $\Norm(\alpha)$.

\head{Outline of lectures}

\begin{enumerate}
\item
Define the absolute norm of an integral ideal. 
\item
Check that the absolute
norm is multiplicative, and that for a principal ideal it equals the absolute
value of the norm of a generator. Define absolute norm for fractional ideals
too.
\item
Note that the number of ideals of bounded absolute norm is finite.
\item
Prove Theorem 5.3: every ideal contains an element which is ``small''
in each embedding (relative to the absolute norm of the ideal).
(This uses Minkowski's lattice point theorem, applied to the Minkowski
space of the number field.)
\item
Conclude that every ideal contains an element of
``small'' norm (Lemma 6.2).
\item
Show that every ideal class contains an ideal of ``small'' absolute norm.
\item
Conclude that the number of ideal classes in a number field $K$
is finite. Define the \emph{class number} of $K$ to be the number of ideal
classes.
\item
Make some comments about class numbers, e.g., of quadratic fields.
\end{enumerate}

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