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\title{Math 725\\Fall 2005\\Homework 3}
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\centerline{\it Due Friday, October 13, 2006.}

\centerline{}

\noindent 1. Let $F\subseteq K$ be fields.
\begin{itemize}
\item[a)] (5 pt) Show that $F+xK[x]$ is always an HFD (note that the analogous result for $F+xK[[x]]$ was done in class).
\item[b)] (5 pt) Show that if $(F+xK[x])[t]$ is an HFD, then $F$ must be algebraically closed in $K$.
\end{itemize}

\centerline{}

\noindent 2. (5 pt) Let $d$ be a square-free integer and $n\in\mathbb{N}$. Consider the domain

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

We will let $\alpha=\sqrt{d}$ if $d\equiv 2,3\text{ mod}(4)$ and $\alpha=\frac{1+\sqrt{d}}{2}$ if $d\equiv 1\text{ mod}(4)$. 

Define the norm map $N:R\longrightarrow\mathbb{Z}$ by

\[
N(a+bn\alpha)=(a+bn\alpha)(a+bn\overline{\alpha})
\]

\noindent where $\overline{\alpha}=-\sqrt{d}$ if $d\equiv 2,3\text{ mod}(4)$ and $\overline{\alpha}=\frac{1-\sqrt{d}}{2}$ if $d\equiv 1\text{ mod}(4)$. Prove the following properties of the norm map.

\begin{itemize}
\item[a)] (5 pt) $N(x)=0$ if and only if $x=0$.
\item[b)] (5 pt) $N(xy)=N(x)N(y)$.
\item[c)] (5 pt) $N(x)=\pm 1$ if and only if $x$ is a unit in $R$.
\item[d)] (5 pt) If $N(x)$ is prime, then $x$ is irreducible in $R$ (does the converse hold?).
\end{itemize}
 
\centerline{}

\noindent 3. (5 pt) Find all nonprime irreducibles in $\mathbb{Z}[\sqrt{-3}]$ (you will find, in fact that this domain is extremely ``close" to being a UFD).

\centerline{}

\noindent 4. (5 pt) Consider the ring of integers $R:=\mathbb{Z}[\sqrt{-89}]$. Find an element of $R$ that has one irreducible factorization of length 2 and another of length 12.



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