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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 6.3  \\ Fall 2019 \hfill Due October 23}}}

\large

\vspace{1cm}

\begin{exercise}
Suppose $\alpha \in \C$ is a nonreal root of $X^2 + pX +q \in \Z[X]$, so that $\Delta = p^2 - 4q <0$.  In class we showed that for $z = a+ b \alpha \in Z[\alpha]$, one has $N(z) = |z|^2 = a^2 - pab + qb^2 \in \N_0$.

\begin{itemize}

\item[\bf a.]  By considering $\Delta$ modulo 4, show that $\Delta \le -3$.

\item[\bf b.]  When $\Delta = -3$, show that $\Z[\alpha] = \Z \left[ \frac{1+\sqrt{-3}}{2} \right]$.  When $\Delta = -4$, show that $\Z[\alpha] = \Z[i]$.  The unit groups of these quadratic orders were computed in Assignment 6.1.

\item[\bf c.]  Show that $z \in \Z[\alpha]^{\times}$ if and only if $N(z) = 1$.

\item[\bf d.]  Show that $N(a+b\alpha) = 1$ if and only if $(2a-bp)^2 + |\Delta| b^2 = 4$.

\item[\bf e.]  When $\Delta \le -5$, show that $\Z[\alpha]^{\times} = \{ \pm 1 \}$.

\end{itemize}

\noindent So, with only two exceptions, the unit group in $\Z[\alpha]$ contains only $\pm 1$.

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\begin{exercise}
Let $\Z[\alpha]$ and $N(\cdot)$ be as above.  Show that for any $z \in \Z[\alpha]$, if $N(z)$ is a prime integer, then $z$ is irreducible.
\end{exercise}


\begin{exercise}
Use the norm and Exercise 1 to prove that $3$ and $2 \pm \sqrt{-5}$ are nonassociate irreducibles in $\Z[\sqrt{-5}]$.
\end{exercise}

\begin{exercise}
Let $D$ be a domain.  Prove that if $a_0 \in D$ is irreducible, then $f = \sum_{i \ge 0} a_i X^i \in D[[X]]$ is irreducible.  Show that this statement is false if $D[[X]]$ is replaced by $D[X]$.
\end{exercise}



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