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\title{Exercise for Section 1:\\The Analytic Class Number Formula}
\author{Math 582e, Winter 2009, University of Washington}
\date{Due Wednesday January 14, 2009}
\begin{document}
\maketitle
\begin{enumerate}
\item A field $K$ is called {\em totally real} if $r_2 = 0$.
Prove that if $K$ is totally real then $\#(U_K)_{\tor} = 2$. Is
the converse true?
\item Prove that complex $s\gg 0$,
$$
\prod_{\text{primes $\p$ in $\O_K$}} \frac{1}{1-N(\p)^{-s}}
\,\, =\,\, \sum_{\text{ideals $I\neq 0$ of $\O_K$}}
\frac{1}{N(I)^{s}}.
$$
\item Use a computer to compute every quantity in the class number
formula for each of the following fields:
$\QQ(\sqrt{5})$, $\QQ(\zeta_5)$, $\QQ(\sqrt[5]{11})$.
\item List all prime ideals of residue characteristic $\leq 7$ in
the fields $\QQ(\sqrt{5})$ and $\QQ(\zeta_5)$.
\item Prove that Dirichlet's Tnit Theorem about the structure
of $U_K^*$ implies that for any squarefree $d\geq 2$, {\em Pell's equation}
$$
x^2 - dy^2 = 1
$$
has infinitely many integer solutions $(x,y)$.
\item Use the functional equation for $\zeta_K(s)$ to prove that the
two formulations of the analytic class number formula in terms of
$\zeta_K^*(1)$ and $\zeta_K^*(0)$ are equivalent. (Hint: See
Theorem 4.9.12 of Cohen's GTM 138).
\end{enumerate}
\end{document}