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\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Lectures 11-12: Dirichlet's Units Theorem \\
September 21 and 24, 2001
\end{center}

\head{Reference}
Neukirch, Sections 1.5 and 1.7.

\head{Outline of lectures}

\begin{enumerate}
\item
State Dirichlet's Units Theorem: the group of units of $\gotho_K$
is isomorphic to the finite group of roots of unity times a free abelian
group of rank $r+s-1$, where $(r,s)$ is the signature of $K$.
\item
Examples: an imaginary quadratic field has finitely many units.
A real quadratic field has a rank 1 group of units; point out relationship
to Pell's equation (see also homework).
\item
Define the logarithm map $K^* \to \prod_{\rho} \RR \times \prod_{\sigma} \RR$,
which on the factor corresponding to an embedding $\tau$ carries
$\alpha$ to $\log |\tau(\alpha)|$. If $K$ has signature $(r,s)$, the image
is a real vector space of dimension $r+s-1$.
\item
Observe that the kernel of the logarithm map on $\gotho_K$ consists of
algebraic integers all of whose conjugates in $\CC$ lie on the unit circle. 
We have seen that there are finitely many of these in $K$, so they 
are roots of unity.
\item
Note that the units of $\gotho_K$ (modulo the roots of unity in $K$)
map into the subspace $H$ of the 
image of the logarithm map consisting of points with sum of coordinates 0.
More precisely, the sum of coordinates of the image of $\alpha$ is
$\log |\Norm(\alpha)|$.
\item
Show that the image of the units of $\gotho_K$ is a discrete subgroup of
$H$, i.e., a subgroup generated by at most $r+s-1$ independent vectors in $H$.
\item
Show that the image of the units of $\gotho_K$ is a \emph{complete} lattice
in $H$, by showing that every element of $H$ can be translated by a unit
into a bounded region. In fact, we'll do this not in $H$ itself, but
its preimage in Minkowski space, the ``norm one surface'', using Minkowski's
lattice point theorem on the integers in Minkowski space.
\item
Deduce the Units Theorem, using the classification of finitely generated
abelian groups.
\item
Note: the space $H$ inherits a norm from the usual norm on $\RR^{r+s}$,
so we can define the volume of the
lattice of units, called the \emph{regulator}. We will not need this for
the moment (but it's closely related to the zeta function of a number field).
\end{enumerate}

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