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\title{Exercises for Section 4.5:\\Computing the Class Group}
\author{Math 582e, Winter 2009, University of Washington}
\date{\bf Due Wednesday February 11, 2009}
\begin{document}
\maketitle
\begin{enumerate}
\item Explain as much as you can about what ``goes on'' when using
the class group algorithm from class to compute the class groups
of the following field:
\begin{enumerate}
\item $x^4 + 6$
\item $x^4 -3x + 5$
\item $x^4-3x-5$
\item $x^3 + 113$
\end{enumerate}
In particular,
\begin{enumerate}
\item What bound $B$ do you use?
\item What is $\#T$?
\item What is the rank of $\OO_K^*$ and $\OO_{K,T}^*$?
\item Do the units of the form $p$ and $\alpha-k$, for $f(k)$ smooth
with $|k|<100$, suffice to generate $\OO_{K,T}^*$ (up to torsion)?
\item If the answer to the previous question is no, how many more
units did you need to throw in using LLL?
\item What are explicit generators for the class group?
\end{enumerate}
You may use Sage as much as you want for this problem, along with the
code handouts at \verb|http://wiki.wstein.org/09/582e/code| (see the
tex files for cut and pastable code). If you want, you can use the
output of Sage/PARI's {\tt class\_number} command to find out what the
true class number is; i.e., you don't have to use the class number
formula to find $\Reg_K \cdot \#\Cl(K)$, and you don't have to compute
an approximation to $\Reg_K$ explicitly... unless you want to.
\end{enumerate}
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