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\title{Math 129: Algebraic Number Theory\\
Take-Home Final Examination}
\author{William Stein}
\date{Due: Monday, May 24, 2004 at 5pm}
\begin{document}
\maketitle

\noindent{}{\bf Remarks:}\\The exam is due on Monday May 24 at 5pm.  You
can turn it in either to my office (under the door) in print
(preferred) or via email as a dvi file.  This exam is worth 20\% of
your grade.  I'm giving it to you well in advance of the due date so
you have more flexibility relative to your schedule in choosing when
to work on it.  \vspace{2ex}

\noindent{}{\bf Rules:}\\You are allowed to use a wide range of
resources on this final exam, including the library, the internet, and
computers.  However, you can not collaborate with anybody else on the
exam (no email or online chatting).  If you do, it's cheating, and
you'll have some bad luck, so don't risk it.  If you find a mistake
or have a question about a problem, email me and I'll post my answer
to the virtual office hours page.\vspace{2ex}

\noindent{}{\bf The 7 Problems:} (each of equal weight)
\begin{enumerate}
\item
\begin{enumerate}
\item Define an analogue of Smith normal form for square matrices with
  entries in  the ring $\Z[i]$ of integers of $\Q(\sqrt{-1})$.
\item Illustrate your answer to (a) by finding Smith normal form of
  the matrix
$$
\mthree{2i}{1+i}{2-i}{i}{2-3i}{3}{1}{-3i}{5}.
$$
\item Does it make sense to define Smith normal form for
squares matrices with entries in the ring of integers $\O_K$
of a general number field?
\end{enumerate}

\item
\begin{enumerate}
\item Prove that there exists an algorithm that computes the group of
  units of a number field.  (Remarks: It was clear that class groups are
  computable from the proof I gave of finiteness of class groups, but
  the proof I gave of Dirichlet's unit theorem did not show that the
  unit group can be computed in practice.  Explain why it can in
  fact be computed.    Remember that for computability speed doesn't
matter---what matters is that there is a finite procedure that is 
guaranteed to terminate with the answer.  Solutions of this problem
can be found in books and probably online, and I encourage you to
look them up and explain what you find.)  
\item Compute the group of units of the cubic field $\Q(\sqrt[3]{2})$.
  Do more than just get the result from a few MAGMA commands;
  instead, use this as an example to illustrate your answer to the first
  part of the problem.
\end{enumerate}

\item Use Hensel's lemma (which we didn't prove in class, but which
  you may assume) to prove that for every integer~$n$ the equation
  $$3x^3+4y^3+5z^3\con 0\pmod{n}$$ has a solution besides the trivial
  solution $x\con y\con z\con 0\pmod{n}$.

\item Let $\rho=\zeta_n + \zeta_n^{-1}$, where $n$ is a primitive
$n$th root of unity.  
\begin{enumerate}
\item 
What additional condition on $n$ is needed for
the ring of integers of $\Q(\rho)$ to be $\Z[\rho]$? 
\item Prove that
your answer is correct.  (Note: Getting the right answer in (a)
without proof in (b) gives you 50\% on this problem.)
\end{enumerate}

\item Suppose $K$ is a number field, $\O_K$ is the ring of integers
  of~$K$, and $\a$ is an ideal of $\O_K$.  Prove that there is a
  finite extension $L$ of $K$ such that the ideal $\a\O_L$ of $\O_L$ is
  principal.

\item Compute the normalized Haar measure of $\AA_K^+/K^+$ for
$K=\Q$ and $K=\Q(i)$.

\item With some work we proved the following crucial lemma (see Lemma
  20.4.1 in the book I made from the course notes, which is available
at the course web page): \setcounter{section}{1}
\begin{lemma}
Let $K$ be a global field.
There is a constant $C>0$ that depends only on~$K$ 
with the following property:

Whenever $\bx=\{x_v\}_v \in \AA_K$ is such that 
\begin{equation}\label{eqn:bignorm}
\prod_v \abs{x_v}_v > C,
\end{equation}
then there is a nonzero principal adele $a \in K\subset \AA_K$ such
that
$$
  \abs{a}_v \leq \abs{x_v}_v \qquad\text{\rm for all $v$}.
$$
\end{lemma}

Problem: Find a value of $C$ that works for $K=\Q$.  Then 
find an $a\in \Q$ as in the conclusion of the lemma for the adele $x=\{x_v\}_v$
with $$x_\infty = 100,\, x_2 = 8,
\, x_3 = 18,\, \text{ and } x_p = 1, \text{ for all }p\geq 5.$$



\end{enumerate}
\end{document}
