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\title{Math 129: Algebraic Number Theory\\
Homework Assignment 2}
\author{William Stein}
\date{Due: Thursday, February 26, 2004}
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\maketitle

\noindent{\bf The Problems:}
\begin{enumerate}

\item 
\begin{enumerate}
\item Let $k$ be a field. Prove that $k[x]$ is a Dedekind domain.
\item (Problem 1.12 from Swinnerton-Dyer)  Let $x$ be an indeterminate.
Show that the ring $\Z[x]$ is Noetherian and integrally closed in its
field of fractions, but is not a Dedekind domain.
\end{enumerate}

\item Use MAGMA to write each of the following (fractional) ideals as a
product of explicitly given prime ideals:
\begin{enumerate}
\item The ideal $(2004)$ in $\Q(\sqrt{-1})$.
\item The ideals $I=(7)$ and $J=(3)$ in the ring of integers of
$\Q(\zeta_7)$, where $\zeta_7$ is a root of the irreducible polynomial
$x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$.    (The field $\Q(\zeta_7)$
is called the $7$th cyclotomic field.)
\item The principal fractional ideal $(3/8)$ in $\Q(\sqrt{5})$.
\end{enumerate}

\item Suppose~$R$ is an order in the ring $\O_K$ of integers of a
number field.  (Recall that an order is a subring of finite index in
$\O_K$.)  For each of the following questions, either explain why the
answer is yes for any possible order $R$ in any $\O_K$, or find one
specific counterexample:
\begin{enumerate}
\item Is $R$ necessarily Noetherian? 
\item Is $R$ necessarily integrally closed in its field of fractions?
\item Is every nonzero prime ideal of $R$ necessarily maximal?
\item Is it always possible to write every ideal of $R$ uniquely as a
product of prime ideals of $R$?
\end{enumerate}

\item Let $\O_K$ be the ring of integers of a number field~$K$.  Prove
that the group of fractional ideals of $\O_K$, under multiplication is
(non-canonically) isomorphic to the group of positive rational numbers
under multiplication.

\end{enumerate}


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