# Math 581f homework 3

\documentclass{article} \include{macros} \begin{document} \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} \end{document}