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\pagestyle{fancy} \lhead{{\sf Homework 14 $|$ Due June 18 (Thursday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Summer 2020 $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-4mm}

\begin{enumerate}


%\item For each of the following ideals, determine if it is prime and
%  if it is maximal.
%  \begin{enumerate}
%  \item The ideal $I=(x)$ in the polynomial ring $R=\Z[x]$.
%  \item The ideal $I=(x)$ in the polynomial ring $R=\R[x]$.
%  \item The ideal $I=(x,y)$ in the multivariate polynomial ring
%    $R=\Z[x,y]$.
%  \item The ideal $I=(x,y)$ in the multivariate polynomial ring
%    $R=\R[x,y]$. \bigskip
%  \end{enumerate}


\item Let $p\in\N$ be a fixed prime. For each of the three ideals
  $I=(p)$, $(x)$, and $(x,p)$ in the ring $R=\Z[x]$, do the
  following steps:
  \begin{enumerate}[(i)]
    \item Describe the elements of the ideal formally, as
      $I=\{\qquad:\qquad\}$.
    \item Characterize the polynomials in $I$ in plain English.
    \item Determine whether $I$ is maximal and/or prime. 
    \item Describe the quotient ring $R/I$. 
  \end{enumerate}
  Then, repeat the above steps for these ideals but in the ring
  $\Q[x]$. \bigskip



\item Let $R$ be a commutative ring with $1$.
  \begin{enumerate}
  \item Prove that $R$ is an integral domain if and only if $0$ is a
    prime ideal.
  \item Prove that an ideal $P\subseteq R$ is prime if and only if
    $R/P$ is an integral domain. 
  \item Show that every maximal ideal is prime. 
  \item Find the group of units $U(R)$ and the maximal ideal(s) of the ring
    \[
    R=\Big\{\frac{a}{b}:a,b\in\Z,\;
    \gcd(a,b)=1,\;p\nmid b\Big\},
    \]
    where $p$ is a fixed prime number. \bigskip
  \end{enumerate}


%\item Recall that $a,b\in R$ are \emph{associates}, denoted $a\sim b$,
%  if $a\mid b$ and $b\mid a$. Show that $a\sim b$ if and only if
%  $a=bu$ for some unit $u\in R$. \bigskip



\item Let $R$ be a principal ideal domain (PID). A \emph{common
  multiple} of $a,b\in R^*$ is an element $m$ such that $a\mid m$ and
  $b\mid m$. Moreover, $m$ is a \emph{least common multiple} (LCM) if
  $m\mid n$ for any other common multiple $n$ of $a$ and $b$.
\begin{enumerate}
\item Prove that any $a,b\in R^*$ have an LCM.
\item Prove that an LCM of $a$ and $b$ is unique up to multiplication
  of associates, and can be characterized as a generator of the
  (principal) ideal $I:=(a)\cap (b)$. \bigskip
\end{enumerate}



\item For any $x=r+s\sqrt{m}\in\Q(\sqrt{m})$, define the \emph{norm} of $x$ to be $N(x)=r^2-ms^2$. %Prove the following about the algebraic integers.
\begin{enumerate}
\item Show that $N(xy)=N(x)N(y)$.
\item Show that $N(x)\in\Z$ if $x\in R_m$.
\item Show that $u\in U(R_m)$ if and only if $|N(u)|=1$. 
\item Show that $U(R_{-1})=\{\pm 1,\pm i\}$, $U(R_{-3})=\{\pm 1,\pm(1\pm\sqrt{3})/2\}$, and $U(R_m)=\{\pm 1\}$ for all other negative square-free $m\in\Z$. \bigskip
\end{enumerate}



%\item Let $R=\Z_{10}$ and $D=\{0,2,4,6,8\}\subset R$. Find the field of fractions of $D$ in $R$. 

%\item Show that $\Q(\sqrt{m})$ is the field of fractions for $R_m$. 


\end{enumerate}


\end{document}

