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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#8} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Nov. $13^\mathrm{th}$, 2020}

\vspace{0.2in}

\begin{enumerate}
\item In each of the following rings, $R$, state the characteristic of the ring. If $R$ has unity, 
     give an example of a unit and its inverse (other than $1$ itself). If no unit exists, explain why not. 
     If $R$ has zero divisors, give an example of a zero divisor. If no zero divisors exist, explain why not.
     \begin{enumerate}[(a)]
     \item $ R = \mathbb{Z}_9 \times \mathbb{Z}_{12} $
     \item $ R = 4\mathbb{Z} = \{ 4k \,|\, k \in \mathbb{Z} \} $ (multiples of 4)
     \item $ R = (\mathbb{Z}_5)^{2 \times 2} $ ($2 \times 2$ matrices with entries in $ \mathbb{Z}_5$)
     \end{enumerate}     
\item Recall $ R \times S $ is the direct product of the rings $ R $ and $ S $.
     \begin{enumerate}[(a)]
     \item Suppose $ R $ and $ S $ have 1's. Then show $ R \times S $ is also a ring with 1.
     \item Let $R$ and $S$ be non-trivial rings (i.e. not the zero ring). Show that $R \times S$ is never an integral domain.
     \end{enumerate}
\item Let $R = \mathbb{Z}[\sqrt{-3}] = \{ a+b\sqrt{-3} \;|\; a,b \in \mathbb{Z} \}$. 

We define the {\it norm} of $z=a+b\sqrt{-3} \in R$ by $N(z) = z\bar{z} = (a+b\sqrt{-3})(a-b\sqrt{-3}) = a^2+3b^2$. Note: $N(z) \in \mathbb{Z}_{\geq 0}$.

   \begin{enumerate}[(a)]
   \item For all $z,w \in R$, prove that $N(zw) = N(z)N(w)$.
   \item When is $N(z)=0$? When is $N(z)=1$?
   \item If $u \in U(R)$, then what can be said about $N(u)$? [{\it Hint:} Consider $N(uu^{-1})$.] Determine $U(R)$.
   \item Prove that $R$ is an integral domain. [{\it Hint:} Use the subring test to show $R$ is a subring of $\mathbb{C}$. Also, although there are other ways to prove this, please use the norm to show $R$ has no zero divisors.]
   \end{enumerate}
\item Prove that $\mathbb{Q}[\sqrt{2}] = \{ a+b\sqrt{2} \;|\; a,b \in \mathbb{Q} \}$ is a field. [Use a subring test to show it is a ring. Use a ``conjugate trick'' to compute the inverse of a non-zero element.]

 \end{enumerate}
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