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\title{Math 129: Algebraic Number Theory\\
Homework Assignment 1}
\author{William Stein}
\date{Due: Thursday, February 19, 2004}
\begin{document}
\maketitle

\noindent{\bf Notes:}
\begin{itemize}
\item Unless otherwise noted, if you can figure out how to use a
computer program to solve a problem, please do.  For complete credit
you must describe exactly how you used the computer (commands typed,
output, etc.)  You might find
\verb| http://modular.fas.harvard.edu/calc/ | useful.
\item You are allowed to work with other people on homework problems,
but you must acknowledge their assistance. 
\item Copying a homework solution if you find it in a book is allowed,
but you must reword it in your own way and {\em cite your sources}.
Learning to use the literature is valuable.
\item If you have questions, email me at {\tt was@math.harvard.edu}.
\end{itemize}

\noindent{\bf The problems:}

\begin{enumerate}

\item %(Artin, problem 12.4.3) 
Let $A=\left(
        \begin{matrix}4&7&2\\2&4&6\\0&0&0
        \end{matrix}\right)$. 
\begin{enumerate}
\item Find invertible integer matrices $P$ and $Q$ such
that $PAQ$ is in Smith normal form.  
\item What is the group structure of
the cokernel of the map $\Z^3\to \Z^3$ defined by multiplication by
$A$?
\end{enumerate}

\item Let $G$ be the abelian group generated by $x,y,z$ with relatoins
$2x+y=0$ and $x-y+3z=0$.  Find a product of cyclic groups that is
isomorphic to $G$.

\item Prove that each of the following rings have infinitely
many prime ideals:
\begin{enumerate}
\item The integers $\Z$.  [Hint: Euclid gave a famous proof of this long ago.]
\item The ring $\Q[x]$ of polynomials over $\Q$.
\item The ring $\Z[x]$ of polynomials over $\Z$.
\item The ring $\Zbar$ of all algebraic integers. [Hint: Use Zorn's
lemma, which implies that every ideal is contained in a maximal
ideal. See, e.g., Prop 1.12 on page 589 of Artin's {\em Algebra}.]
\end{enumerate}

\item 
(This problem was on the graduate qualifying exam on Tuesday.)  Let
$\Zbar$ denote the subset of all elements of $\Qbar$ that satisfy a
monic polynomial with coefficients in the ring $\Z$ of integers.  We
proved in class that $\Zbar$ is a ring.
\begin{enumerate}
\item Show that the ideals $(2)$ and $(\sqrt{2})$ in $\Zbar$ are
distinct.
\item Prove that $\Zbar$ is not Noetherian.
\end{enumerate}

\item% (1.2 from Swinnerton-Dyer): 
Show that neither $\Z[\sqrt{-6}]$
nor $\Z[\sqrt{5}]$ is a unique factorization domain.  [Hint: Consider
the factorization into irreducible elements of $6$ in the first case
and~$4$ in the second.  A nonzero element $a$ in a ring $R$ is an {\em
irreducible element} if it is not a unit and if whenever $a=qr$, then
one of $q$ or $r$ is a unit.]

\item Find the ring of integers of each of the following number
fields:
\begin{enumerate}
\item $\Q(\sqrt{-3})$,
\item $\Q(\sqrt{3})$, and
\item $\Q(\sqrt[3]{2})$.
\end{enumerate}
Do not use a computer for the first two.

\item Find the discriminants of the rings of integers of the numbers
fields in the previous problem.  (Do not use a computer.)

\item Let $R$ be a finite integral domain.  Prove that $R$ is a field. 
[Hint: Show that if $x$ is a nonzero element, then $x$ has an inverse
by considering powers of $x$.]

\item Suppose $K\subset L\subset M$ is a tower of number fields
and let $\sigma:L\hra \Qbar$ be a field embedding of $L$ into $\Qbar$
that fixes $K$ elementwise.  Show that $\sigma$ extends in exactly
$[M:L]$ ways to a field embedding $M\hra\Qbar$.

\item
\begin{enumerate}
\item  Suppose $I$ and $J$ are principal ideals in a ring $R$.
Show that the set $\{ab : a \in I,\, b\in J\}$ is an ideal.  
\item Give an example of ideals $I$ and $J$ in the polynomial
ring $\Q[x,y]$ in two variables such that 
$\{ab : a \in I,\, b\in J\}$ is not an ideal.
Your example illustrates why it is necessary to define the product
of two ideals to be the ideal generated by
$\{ab : a \in I,\, b\in J\}$.
\item Give an example of a ring of integers $\O_K$ of a number
field, and ideals~$I$ and~$J$ such that $\{ab : a \in I,\, b\in J\}$
is not an ideal.
\end{enumerate}

\end{enumerate}


\end{document}