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ant07/homework/hw1

Homework 1, due Wednesday October 10, 2007

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

Homework 1 PDF

\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}


2013-05-11 18:33