\documentclass[11pt]{article}\usepackage[active]{srcltx}
\voffset=-0.05\textheight
\textheight=1.1\textheight
\input{macros}

\title{Math 129: Algebraic Number Theory\\
Homework Assignment 1}
\author{William Stein}
\date{\bf Due: Thursday, February 17, 2005}
\begin{document}
\maketitle

\noindent{\em The problems have equal point value,
and multi-part problems are of the same value.  There
are $4$ problems.}


\begin{enumerate}

\item 
Let $A=\left(
        \begin{matrix}1&2&3\\4&5&6\\7&8&9
        \end{matrix}\right)$. 
\begin{enumerate}
\item Find the Smith normal form of $A$.
\item Prove that 
the cokernel of the map $\Z^3\to \Z^3$ given by multiplication by~$A$ 
is isomorphic to $\Z/3\Z \oplus \Z$.
\end{enumerate}

%\item Give an example of a ring $R$ that is not noetherian. 

\item Show that the minimal polynomial of an algebraic number
  $\alpha\in\Qbar$ is unique.  (You may assume that $\Q[x]$ is a
  prinicipal ideal domain, which is easy to prove using the division
  algorithm for polynomials.)

\item Which of the following rings have infinitely
many prime ideals?  (Prove that your answers are correct.)
\begin{enumerate}
\item The integers $\Z$. 
\item The ring $\Z[x]$ of polynomials over $\Z$.
\item The quotient ring $\C[x]/(x^{2005}-1)$.
\item The ring $(\Z/6\Z)[x]$ of polynomials over the ring $\Z/6Z$.
\item The quotient ring $\Z/n\Z$, for a fixed positive integer~$n$.
\item The rational numbers~$\Q$.
\item The polynomial ring $\Q[x,y,z]$ in three variables.
\end{enumerate}

\item Which of the following numbers are algebraic integers?
\begin{enumerate}
\item The number $(1+\sqrt{5})/2$.
\item The number $(2+\sqrt{5})/2$.
\item The value of the infinite sum $\sum_{n=1}^{\infty} 1/n^2$.
\item The number $\alpha/3$, where $\alpha$ is a root of
$x^4 + 54x + 243$.
\end{enumerate}

\item Prove that $\Zbar$ is not noetherian.

\end{enumerate}
\end{document}
%%% Local Variables: 
%%% mode: latex
%%% TeX-master: t
%%% End: 
