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\def\classname{Math 3140 --- Fall 2012}

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\begin{exercise}
An element of a ring $R$ is called a \textbf{unit} if it has a multiplicative inverse.  The set of all units of $R$ is denoted $R^\ast$.
\begin{enumerate}[label=(\alph{*})]
\item Let $\bR[x]$ be the ring of polynomials with real coefficients.  What is $\bR[x]^\ast$?
\item Let $R = \bR[i] / (i^2 = -1) = \bC$.  What is $R^\ast$?
\item Let $R = \bR[\epsilon] / (\epsilon^2 = 0)$ be the ring of symbols $a + b \epsilon$ with
\begin{gather*}
(a + b \epsilon) + (a' + b' \epsilon) + (a + a') + (b + b') \epsilon \\
(a + b \epsilon)(a' + b' \epsilon) = a a' + (a b' + a' b) \epsilon .
\end{gather*}
What is $R^\ast$?
\item Let $R = \bR[x] / (x^2 = 1)$ be the ring of symbols $a + bx$ with operations
\begin{equation*}
(a + bx) + (a' + b'x) = (a + a') + (b + b') x \\
(a + bx)(a' + b'x) = (a a' + b b') + (a b' + a' b) x .
\end{equation*}
What is $R^\ast$?
\end{enumerate}
\end{exercise}

\vskip .5cm

\noindent The ring of \textbf{Gaussian integers} is the set of all complex numbers $x + iy$ where $x, y \in \bZ$.  The Gaussian integers are denoted $\bZ[i]$.  The addition and multiplication rules are just as in the complex numbers:
\begin{gather*}
(x + iy) + (x' + iy') = (x + x') + i(y + y') \\
(x + iy)(x' + iy') = x x' - y y' + i(x y' + x' y) .
\end{gather*}

\begin{exercise}
An element of a ring is called a \textbf{unit} if it has a multiplicative inverse.
\begin{enumerate}[label=(\alph{*})]
\item List of all units of $\bZ[i]$.
\item Can you prove that your list is complete?  
\end{enumerate}
\end{exercise}

\begin{exercise}
An element $x$ of a commutative ring $R$ is called \textbf{irreducible} if it cannot be written as $yz$ unless $y$ or $z$ is a unit.
\begin{enumerate}[label=(\alph{*})]
\item What are the irreducible elements of $\bZ$?
\item Is $2 = 2 + 0 i$ irreducible in $\bZ[i]$?
\item What about $3$?
\item Make a list of some irreducible elements of $\bZ$ and determine which are irreducible in $\bZ[i]$.
\item Make a conjecture about which irreducible elements of $\bZ$ remain irreducible in $\bZ[i]$.
\end{enumerate}
\end{exercise}

\vskip .5cm

\begin{exercise}
Hint:  it might be helpful to think about both parts of this exercise at the same time.
\begin{enumerate}[label=(\alph{*})]
\item Let $\bC[x]$ be the ring of polynomials with  complex coefficients.  What are the irreducible elements of $\bC[x]$?
\item Let $\bR[x]$ be the ring of polynomials with real coefficients.  What are the irreducible elements of $\bR[x]$?
\end{enumerate}
\end{exercise}

\vskip.5cm

\noindent The ring of \textbf{Eisenstein integers} is the set of all complex numbers $x + \Bigl( \frac{1}{2} + i \frac{\sqrt{3}}{2} \Bigr) y$ where $x, y \in \bZ$.  It is sometimes denoted $\bZ[\rho]$ where $\rho = \frac{1}{2} + i \frac{\sqrt{3}}{2}$.  However, the notation $\bZ[\rho]$ is much less standard than $\bZ[i]$; the name ``Eisenstein integers'' is also not used as widely as ``Gaussian integers'' is.  Addition and multiplication in the Eisenstein integers are again as in the complex numbers.

\begin{exercise}
Let $\rho = \frac{1}{2} + i \frac{\sqrt{3}}{2}$ as above.
\begin{enumerate}[label=(\alph{*})]
\item Remember that we really should check that $\bZ[\rho]$ is a ring.  We won't do this now because we'll be able to do it more efficiently later.  
\item Compute $\rho^2$ and $\rho^3$ and verify that these are Eisenstein integers.
\item Make a list of all the units in $\bZ[\rho]$.
\item Which irreducible elements of $\bZ$ remain irreducible in $\bZ[\rho]$?
\end{enumerate}
\end{exercise}

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