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\section*{Math 403/503 Spring 2024}
In this class, by
\begin{itemize}
\item a \emph{ring} $R$ we mean a \emph{ring with identity $1_R$};
\item a \emph{subring} $S\subseteq R$ we mean \emph{subring containing the identity $1_R$};
\item a \emph{ring homomorphism} $\varphi:R\to T$ we mean \emph{ring homomorphism sending $1_R$ to $1_T$}.
\end{itemize}
\subsection*{Homework 7, due March 27}
\begin{enumerate}
\item The ring of \emph{Gaussian integers}, $\mathbb{Z}[i]=\{a+bi\in\mathbb{C}\mid a,b\in\mathbb{Z}\}$, is a UFD. Factor each of the following elements in $\mathbb{Z}[i]$ into a product of irreducibles.
\begin{enumerate}[{\rm (a)}]
\item $5$
\item $6+8i$
\item $1+3i$
\end{enumerate}
\item Prove that the field of fractions of the Gaussian integers, $\mathbb{Z}[i]$, is isomorphic to \[\mathbb{Q}(i) = \{p + qi : p, q \in\mathbb{Q}\}.\]
\smallskip
\item Let $D$ be an integral domain. Define a relation on $D$ by $a\sim b$ iff $a$ and $b$ are associates in $D$. Prove that $\sim$ is an equivalence relation on $D$.
\smallskip
\item An ideal $I$ of a commutative ring $R$ is said to be \emph{finitely generated} if there exist elements $a_1,a_2,\ldots,a_n$ in $R$ such that every element $r$ in $I$ can be written $r=a_1r_1+\cdots+a_nr_n$
for some $r_1,\ldots,r_n$ in $R$. Prove that $R$ satisfies the ascending chain condition if and only if every ideal of $R$ is finitely generated.
\smallskip
\item Show that $\mathbb{Z}[\sqrt{-5}]$ is not a unique factorization domain.
\smallskip
\item Prove that $\Z[x]$ cannot be a Euclidean domain. 
\smallskip
\item Let $D$ be a Euclidean domain with Euclidean valuation $\nu$. If $a$ and $b$ are associates in $D$, prove that $\nu(a) = \nu(b)$.
\end{enumerate}
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