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\title{Math 724\\Fall 2010\\Homework 2}
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\centerline{\it Due Friday, February 19, 2010.}

\centerline{}

\noindent 1. Let $R$ be an integral domain. A nonzero nonunit element $z\in R$ is said to be a {\it universal side divisor} if given any $x\in R$ there is a $r\in R$ such that 

\[
x=rz+v
\]

\noindent where $v$ is either 0 or a unit in $R$. Let $R$ be a Euclidean domain with norm function $\phi$. 
\begin{itemize}
\item[a)] (5 pt) Show that any nonunit in $R$ of minimal norm is a universal side divisor.
\item[b)] (5 pt) Show that $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is not Euclidean.
\end{itemize}

\centerline{}

\noindent 2. Let $d$ be a squarefree integer. We define

\[
R=\mathbb{Z}[\omega] \text{ where } \omega=
\begin{cases}
\sqrt{d}, \text{ if $d\equiv 2,3$ mod($4$);}\\
\frac{1+\sqrt{d}}{2}, \text{ if $d\equiv 1$ mod($4$).}
\end{cases}
\]

\begin{itemize}
\item[a)] (5 pt) Show that $R$ is integral over $\mathbb{Z}$.
\item[b)] (5 pt) We define a {\it norm} to be a map $N:R\longrightarrow\mathbb{N}_0$ satisfying $N(0)=0$ and $N(ab)=N(a)N(b)$. Show that $N:\mathbb{Z}[\omega]\longrightarrow\mathbb{N}_0$ defined by $N(a+b\omega)=(a+b\omega)(a+b\overline{\omega})$ is a norm.
\item[c)] (5 pt) Use the norm to show that $\mathbb{Z}[\omega]$ is atomic.
\item[d)] (5 pt) Show that the ring $\mathbb{Z}[\sqrt{-14}]$ is not a UFD.
\end{itemize}

\centerline{}

\noindent 3. Let $R$ be a domain and $N$ a norm on $R$. We say that $N$ is a {\it Dedekind-Hasse norm} if $N$ is positive and for every nonzero $x,y\in R$ either $y$ is divisible by $x$ or we can find $a,b\in R$ such that 

\[
0<N(ax+by)<N(x).
\]

\begin{itemize}
\item[a)] (5 pt) Show that $R$ is a PID if and only if $R$ has a Dedekind-Hasse norm.
\item[b)] (5 pt) Show that the norm defined in problem 2 for the ring $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a Dedekind-Hasse norm (hence $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a PID that is not Euclidean).
\end{itemize}

\centerline{}

\noindent 4. Suppose that $R$ is a UFD.
\begin{itemize}
\item[a)] (5 pt) Show that $R[[x]]$ is atomic.
\item[b)] (5 pt) Show that if $f(x)\in R[[x]]$ is such that $f(0)=\prod_{i=1}^n p_i^{a_i}$ (with the $p_i$'s distinct nonzero prime elements of $R$ and each $a_i>0$) and $f(x)=\prod_{j=1}^t f_j(x)$ (with each $f_j(x)$ irreducible) then $1\leq t\leq \sum_{i=1}^n a_i$. Give examples to show that both bounds can be achieved.
\item[c)] (5 pt) Suppose that $R$ is a PID. Show that if $f(x)\neq x$ is irreducible in $R[[x]]$ then $f(x)=p^n+xg(x)$ with $p$ a nonzero prime in $R$ and $g(x)\in R[[x]]$ (is the converse true?). 
\item[d)] (5 pt) With the notation as above, show that if $R$ is a PID, then $n\leq t\leq\sum_{i=0}^n a_i$.
\end{itemize}

\centerline{}

\noindent 5. Let $R$ be a domain with quotient field $K$. $\omega\in K$ is called almost integral over $R$ if there is a nonzero $r\in R$ such that $rx^n\in R$ for all $n\geq 0$. If $R$ contains all of the elements $\omega\in K$ that are almost integral over $R$, we say that $R$ is completely integrally closed.
\begin{itemize}
\item[a)] (5 pt) Show that any UFD is completely integrally closed.
\item[b)] (5 pt) Suppose that $A\subseteq B$ are integral domains. Completely characterize when the domain $A+xB[x]$ is a UFD.
\end{itemize}





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