# Homework 1, due Wednesday October 10, 2007

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

\documentclass{article} \include{macros} \begin{document} \begin{center} \Large\bf Homework 1 for Math 581F, Due October 10, 2007\end{center} The above date is not a typo. Each problem has equal weight, and parts of problems are worth the same amount as each other. \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. \item Which of the following rings have infinitely many prime ideals? \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 $\Z/6\Z$. \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}