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\begin{document}
\title{Homework $\#$8}
\author{Due 11/13/13 at start of class }
\maketitle
\subsection*{Problem 1 (D$\&$F 8.1.3)} Let R be a Euclidean Domain. Let $m$ be the minimum integer in the set of norms of nonzero elements of $R$. Prove that every nonzero element of norm $m$ is a unit. Deduce that a nonzero element of norm zero (if such an element exists) is a unit. 

\subsection*{Problem 2 (D$\&$F 8.1.7)} Find a generator for the ideal $(85,1+13i)$ in $\mathbb{Z}[i]=\mathbb{Z}[\sqrt{(-1)}]$. 

\noindent [Hint: $\mathbb{Z}[i]$ is a Euclidean Domain with respect to the norm $N(a+bi)=a^2+b^2$ (cf page 272).]


\subsection*{Problem 3 (D$\&$F 8.1.11,8.2.2)}
Let $R$ be a commutative ring with 1 and let $a$ and $b$ be nonzero elements of $R$.  A \textit{least common multiple} of $a$ and $b$ is an element $e$ of $R$ such that
\begin{itemize}
\item[(i)] $a\mid e$ and $b\mid e$, and
\item[(ii)] if $a\mid e'$ and $b\mid e'$ then $e\mid e'$ 
\end{itemize}
\begin{itemize}
\item[(a)] Prove that a least common multiple of $a$ and $b$ (if such exists) is a generator for the unique largest principal ideal contained in $(a)\cap (b)$.
\item[(b)] Deduce that any two nonzero elements in a Euclidean Domain have a least common multiple which is unique up to multiplication by a unit.
\item[(c)] Prove that in a Euclidean Domain the least common multiple of $a$ and $b$ is $\frac{ab}{(a,b)}$, where $(a,b)$ is the greatest common divisor of $a$ and $b$.
\item[(d)] Prove that any two nonzero elements of a PID have a least common multiple.
\end{itemize}
\subsection*{Problem 4 (D$\&$F 8.2.3)} Let $R$ be an integral domain and suppose that every prime ideal in $R$ is principal. This exercise proves that every ideal of $R$ is principal, i.e. $R$ is a PID.

First let $\mathcal{I}$ be the set of ideals of $R$ that are not principal. Assume that if $\mathcal{I}$ is nonempty then it has a maximal element under inclusion. (Challenge: read about Zorn's lemma and prove this statement).

\begin{itemize}
\item[(a)] Let $I$ be a maximal element of $\mathcal{I}$ and let $a,b\in R$ with $ab\in I$ but $a\notin I$ and $b\notin I$. Let $I_a=(I,a)$ be the ideal generated by $I$ and $a$. Let $I_b=(I,b)$ be the ideal generated by $I$ and $b$, and define $J=\{r\in R\mid rI_a\subseteq I\}$. Prove that $I_a=(\alpha)$ and $J=(\beta)$ are principal ideals in $R$ with $I\subsetneq I_b\subseteq J$ and $I_aJ=(\alpha\beta)\subseteq I$..
\item[(b)] If $x\in I$ show that $x=s\alpha$ for some $s\in J$. Deduce that $I=I_aJ$ is principal, a contradiction, and conclude that $R$ is a PID.
\end{itemize}

\subsection*{Problem 5}  In each of the following rings, give at least two distinct factorizations of some nonzero non-unit into irreducibles.
\begin{itemize}
\item[(a)] $\mathbb{Q}[x^2,x^3]$. [This ring consists of finite sums of elements of the form $a(x^2)^i(x^3)^j$ (see pages 296-297 in D\&F).]
\item[(b)] $\mathbb{Z}[\sqrt{-13}]$.
\end{itemize}

\subsection*{Problem 6 (D$\&$F 9.2.3)} 
Let $F$ be a field and let $f(x)$ be a polynomial in $F[x]$. Prove that $F[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible. 
 
%\subsection*{Problem 7 (D$\&$F 9.2.4)} Let $F$ be a finite field. Prove that $F[x]$ contains infinitely many primes. (Note that over an infinite field the polynomials of degree 1 are an infinite set of primes in the ring of polynomials.)

\subsection*{Problem 7 (D$\&$F 9.2.6)}Describe the ring structure of the following rings:
\begin{itemize}
\item[(a)] $\mathbb{Z}[x]/(2)$.
\item[(b)] $\mathbb{Z}[x]/(x)$.
\item[(c)] $\mathbb{Z}[x]/(x^2)$.

\end{itemize}

\subsection*{Challenge Problem} Determine the characteristics (cf Homework 7 \#4) of each of the rings in Problem 7.


\end{document}