Math 581f homework 3

attachment:hw3.pdf

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\begin{center}
\Large\bf Homework 3 for Math 581F, Due FRIDAY October 19, 2007\end{center}

Each problem has equal weight, and parts of problems are worth the
same amount as each other.


\begin{enumerate}
\item Prove that $\Zbar$ 
is integrally closed in its field of fraction, and every nonzero prime
ideal is maximal.  Thus $\Zbar$ is not a Dedekind domain only because
it is not noetherian.

\item Let $K$ be a field.  
\begin{enumerate}
\item Prove that the polynomial ring $K[x]$
is a Dedekind domain.
\item Is $\Z[x]$ a Dedekind domain?
\end{enumerate}

\item \label{ex:finitedomain} 
Prove that every finite integral domain is a field.

\item \label{ex:idealprod} 
\begin{enumerate}
\item Give an example of two ideals $I, J$ in a
commutative ring $R$ whose product is {\em not} equal to the set
$\{ab : a \in I, b \in J\}$. 
\item Suppose $R$ is a principal ideal domain.   
Is it always the case that 
$$ 
IJ = \{ab : a \in I, b \in J\}
$$
for all ideals $I, J$ in $R$?
\end{enumerate}

\item Is the set $\Z[\frac{1}{2}]$ of rational numbers with
  denominator a power of $2$ a fractional ideal?

\item Suppose you had the choice of the following two jobs\footnote{From {\em The Education of T.C. MITS} (1942).}:
\begin{itemize}
\item[Job 1] Starting with an annual salary of \$1000,
and a \$200 increase every year.
\item[Job 2] Starting with a semiannual salary of \$500,
and an increase of \$50 every 6 months.
\end{itemize}
In all other respects, the two jobs are exactly alike.
Which is the better offer (after the first year)?  
Write a Sage program that creates a table showing how
much money you will receive at the end of each year for
each job. (Of course you could easily do this by hand -- the
point is to get familiar with Sage.)

\item Let $\O_K$ be the ring of integers of a number field.
Let~$F_K$ denote the abelian group of fractional ideals of $\O_K$.
\begin{enumerate}
\item Prove that $F_K$ is torsion free.
\item Prove that $F_K$ is not finitely generated.
\item Prove that $F_K$ is countable.
\item Conclude that if $K$ and $L$ are number fields, then there
exists some (non-canonical) isomorphism of groups $F_K\ncisom F_L$.
\end{enumerate}

\item From basic definitions, find the rings of integers of the fields
$\Q(\sqrt{11})$ and $\Q(\sqrt{-6})$.  Check your answers using Sage.

\item In this problem, you will give an example to illustrate the
  failure of unique factorization in the ring $\O_K$ of integers of
  $\Q(\sqrt{-6})$.
\begin{enumerate}
\item Give an element $\alpha \in \O_K$ that factors in two distinct
  ways into irreducible elements.  
\item Observe explicitly that the $(\alpha)$ factors uniquely, i.e.,
  the two distinct factorization in the previous part of this problem
  do not lead to two distinct factorization of the ideal $(\alpha)$
  into prime ideals.
\end{enumerate}

\item Factor the ideal $(10)$ as a product of primes
in the ring of integers of $\Q(\sqrt{11})$.  You're allowed
to use Sage, as long as you show the commands you use.


\end{enumerate}

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