\documentclass[11pt]{article}
\include{macros}
\usepackage{url}
\title{Exercises for Section 3:\\The Maximal Order}
\author{Math 582e, Winter 2009, University of Washington}
\date{\bf Due Wednesday January 28, 2009 (note: extra week)}
\begin{document}
\maketitle
\begin{enumerate}
\item Prove that if $\O$ is an order in $\O_K$,
then $[\O_K:\O]^2 = \disc(\O)/\disc(\O_K)$.
\item Prove that every order $\O \subset \ZZ[i] = \ZZ[\sqrt{-1}]$
is of the form $\ZZ+fi\ZZ$ for some $f\in\ZZ_{\geq 1}$.
Prove that if $\ZZ + fi\ZZ = \ZZ + gi\ZZ$ for $f,g \in \ZZ_{\geq 1}$,
then $f=g$.
\item Use the {\em naive algorithm} to compute the maximal order
of $K=\QQ(\sqrt{5})$.
\item Prove that the radical of an ideal $I$ in a commutative ring
$R$ is an ideal, where
$$
\text{rad}(I) = \{x \in R : x^j \in I \text{ for some } j > 0\}.
$$
\item Compute a $3$-maximal order in $\QQ(\sqrt[3]{2})$ using the
``round 2'' algorithm from class (but without the Dedekind
termination condition). Explicitly compute each matrix that comes
up in the algorithm, etc., just like in class. You can use a
computer to do the actual linear algebra.
\item Compute $\O_K$ for $K=\QQ(\alpha)$. where $\alpha^3 + \alpha^2 -
2\alpha + 8 = 0$ using the ``round 2'' algorithm from class (but
without the Dedekind termination condition).
Explicitly compute each matrix that comes up in the algorithm, etc.,
just like in class. You can use a computer to do the actual linear
algebra.
\item Create a general implementation of round 2 (without early
termination) to compute a $p$-maximal order in any number field.
Use your program to check your answers for the previous two
problems.
\end{enumerate}
\end{document}