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\begin{center}
  {\large\textbf{Dartmouth College}\\} Mathematics 101\\\medskip
  Homework 7 \small{(due Wednesday, November 28)}
\end{center}

\begin{enumerate}
\item Let $A$ be a commutative ring with identity. Let $I_1, \dots,
  I_n$ be ideals in $A$ which are coprime in pairs.  Show that
  $I_1I_2\cdots I_n = I_1\cap I_2\cap \cdots \cap I_n.$

\item Let $A$ be a commutative ring with identity, and let $X$ be the
  set of all prime ideals in $A$. $X$ is called the prime spectrum of
  $A$, written $Spec(A)$.  For each subset $E\subseteq A$, let $V(E)$
  denote the set of primes ideals of $A$ which contain $E$.  Prove
  that:
  \begin{enumerate}
  \item If $I = \la E\ra$ is the ideal generated by $E$, then $V(I) =
    V(E)$.
  \item Show that $V(0) = X$ and $V(1) = \emptyset$.

  \item If $\{E_i\}_{i\in I}$ is any family of subsets of $A$, then
    $V(\cup_i E_i) = \cap_{i\in I}V(E_i)$.

  \item For any ideals $I,J$ of $A$, show that $V(I\cap J) = V(IJ) =
    V(I) \cup V(J)$.

    These properties demonstrate that the sets $V(E)$ satisfy the
    axioms for closed sets in a topological space.  This topology is
    called the Zariski topology on $Spec(A)$.
  \end{enumerate}

\item Let $A = \Z$ and $\p = p\Z$ with $p$ a prime in $\Z$.  We have
  characterized the localization $A_\p = \Z_\p$ as $\{a/b \in \Q \mid
  a,b \in Z,\ p\nmid b,\ \gcd(a,b) = 1\}$.
  \begin{enumerate}
  \item Characterize the unit group $\Z_\p^\times$.

  \item Show that every nonzero element in $\Z_\p$ can be written
    uniquely as $p^\nu u$ where $\nu$ is a nonnegative integer and $u
    \in \Z_\p^\times$.  You may of course assume unique factorization
    in $\Z$.

  \item Characterize all the ideals of $\Z_\p$, and confirm that
    $\Z_\p$ has a unique maximal ideal.

  \item Show that $\Z_\p/ p\Z_\p \cong \Z/p\Z$.
  \end{enumerate}



\item Consider the localization of $\Z[x]$ at the prime ideal $(x)$.
  \begin{enumerate}
  \item Describe the elements of $\Z[x]_{(x)}$.
  \item Is $(x)$ maximal in $\Z[x]_{(x)}$?  If so, describe the
    resulting quotient field.
  \item How does $\Z[x]_{(x)}$ compare to $\Q[x]_{(x)}$?
  \end{enumerate}
\end{enumerate}

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