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\title{Math 129: Algebraic Number Theory\\
Homework Assignment 4}
\author{William Stein}
\date{Due: Thursday, March 11, 2004}
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\noindent{\bf The problems:}
\begin{enumerate}

\item Find representative ideals for each element of the class group
of $\Q(\sqrt{-23})$.  Illustrate how to use the Minkowski bound
to prove that your list of representatives is complete.

\item Suppose $\O$ is an order in the ring of integers $\O_K$ of a
number field.  Is every ideal in $\O$ necessarily generated by two
elements?

\item Let $K$ be a number field of degree $n>1$ with $s$ pairs
of complex conjugate embeddings.  Prove that 
\[
\left(\frac{\pi}{4}\right)^s\cdot \frac{n^n}{n!} > 1.
\]

\item Do the exercise on page 19 of Swinnerton-Dyer, which shows
that the quantity $C_{r,s}$ in the finiteness of class group theorem
can be taken to be $\left(\frac{4}{\pi}\right)^{s} \frac{n!}{n^n}$.

% exercise from page 341 of Frohlich-Taylor
\item Let $\alpha$ denote a root of $x^3-x+2$ and let $K=\Q(\alpha)$.
Show that $\O_K=\Z[\alpha]$ and that $K$ has class number $1$ (don't
just read this off from the output of the MAGMA commands {\tt
MaximalOrder} and {\tt ClassNumber}).  [Hint: consider the square
factors of the discriminant of $x^3-x+2$ and show that
$\frac{1}{2}(a+b\alpha + c\alpha^2)$ is an algebra integer if and only
if $a$, $b$, and $c$ are all even.]

% from page 342 of Frohlich-Taylor
\item If $S$ is a closed, bounded, convex, symmetric set in $\R^n$
with $\Vol(S)\geq m 2^n$, for some positive integer $m$, show that
$S$ contains at least $2m$ nonzero points in $\Z^n$.

\end{enumerate}


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