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\title{Math 129: Algebraic Number Theory\\
Homework Assignment 5}
\author{William Stein}
\date{Due: Thursday, March 18, 2004}
\begin{document}
\maketitle
\begin{enumerate}

\item Prove that any finite subgroup of the multiplicative
group of a field is cyclic.

\item For a given number field $K$, which seems more difficult for
MAGMA to compute, the class groups or explicit generators for the
group of units?  It is very difficult (but not impossible) to not get
full credit on this problem.  Play around with some examples, see what
seems more difficult, and {\em justify} your response with examples.
(This problem might be annoying to do using the MAGMA web page, since
it kills your MAGMA job after 30 seconds.  Feel free to request a
binary of MAGMA from me, or an account on MECCAH (Mathematics Extreme
Computation Cluster at Harvard).)

\item
\begin{enumerate}
\item Prove that there is no number field $K$ such that $U_K\isom \Z/10\Z$.
\item Is there a number field $K$ such that 
$U_K\isom \Z\times \Z/6\Z$?
\end{enumerate}

\item Prove that the rank of $U_K$ is unbounded as $K$ varies over all
number fields.

\item Let $K=\Q(\zeta_5)$.
\begin{enumerate}
\item Show that $r=0$ and $s=2$.
\item Find explicitly generators for the group of
units of $U_K$ (you can use MAGMA for this).
\item Draw an illustration of the log map
$\vphi:U_K\to \R^2$, including the hyperplane
$x_1+x_2=0$ and the lattice in the hyperplane 
spanned by the image of $U_K$.
\end{enumerate}

\item
Find the group of units of $\Q(\zeta_n)$ as an abstract
group as a function of $n$.  (I.e., find the number of cyclic
factors and the size of the torsion subgroup.  You do not have
to find explicit generators!)

\item Let $K=\Q(a)$, where $a$
is a root $x^3-3x+1$.
\begin{enumerate}
\item Show that $r=3$.
\item Find explicitly the log embedding of $U_K$
into a $2$-dimensional hyperplane in $\R^3$, and
draw a picture.
\end{enumerate}

\item Prove that if $K$ is a quadratic field and the torsion subgroup
of $U_K$ has order bigger than $2$, then $K=\Q(\sqrt{-3})$ or
$K=\Q(\sqrt{-1})$.

\item A {\em Salem number} is a real algebraic integer, greater than
1, with the property that all of its conjugates lie on or within the
unit circle, and at least one conjugate lies on the unit circle.  By
any method (including ``google''), give two examples of Salem numbers.


\end{enumerate}


\end{document}
