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\title{Math 725\\Fall 2005\\Homework 2}
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\centerline{\it Due Friday, September 23, 2006.}

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\noindent 1. 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}

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\noindent 2. Let $R$ be a domain and $f(x)\in R[x]$. We define the content of the polynomial $f$ to be the ideal (in $R$) generated by the coefficients of $f$ (that is, if $f(x)=\sum_{i=0}^na_ix^i$ then $c(f)=(a_0,a_1,\cdots, a_n)$). Show that if $f,g\in R[x]$ then $c(fg)\subseteq c(f)c(g)$ and give an example to show that this containment may be strict.

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\noindent 3. 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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