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\noindent{\Large{\textsc{Number Theory\hfill Assignment 12.2  \\ Fall 2020 \hfill Due November 30}}}

\large

\vspace{1cm}

\begin{exercise}
Given a quaternion $z =a + bi + cj + dk \in \mathbb{H}$ with $a,b,c,d \in \R$, recall that $z^{\ast} = a - bi - cj - dk$.  For $z, w \in \mathbb{H}$ we define their {\em dot product} and {\em cross product} to be
\[
z \cdot w = \frac{wz^{\ast} + zw^{\ast} }{2} \,\,\, \text{ and } \,\,\, z \times w = \frac{wz^{\ast} - zw^{\ast}}{2},
\]
respectively.  
\begin{itemize}

\item[\bf a.]  Prove that $z \cdot z = a^2 + b^2 + c^2 + d^2$.

\item[\bf b.]  Prove that if $z, w \in \mathbb{H}$ are purely imaginary (i.e.\ have no real coordinate), then $z \cdot w$ and $z \times w$ agree with the usual dot and cross products of vectors in $\R^3$.

\end{itemize}

\end{exercise}

\begin{exercise}
Show that if $z \in \mathbb{H}$, then $z^{\ast}z = z z^{\ast} = N(z)$.  Conclude that if $z \ne 0$, then $z^{-1} = z^{\ast}/N(z)$.
\end{exercise}

\begin{exercise}
Show that every nonzero $z \in \mathbb{H}$ has the {\em polar form} $z = \rho (\cos \theta + \sin \theta \mathbf{u})$, where $\theta \in \R$, $\rho = \sqrt{N(z)}$, $\mathbf{u}$ is purely imaginary, and $N(\mathbf{u})=1$.
\end{exercise}

\begin{exercise}
Textbook exercise 13.3.7.
\end{exercise}

\begin{exercise}
Textbook exercise 13.3.13.
\end{exercise}


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