Homework 1, due Wednesday October 10, 2007
Each problem has equal weight, and parts of problems are worth the same amount as each other.
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\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}
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