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\author{Geoff Smith}
\title{Extra questions for Chapter 3}
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\section*{Extra questions for Chapter 3}
\begin{enumerate}
\item
The early parts of this question will help you do the later parts.
Let $w = e^{2\pi i/3}.$
\item[(a)] Show that every complex number
can be uniquely written as $a + bw$ with
$a, b$ real.
\item[(b)] Show that $w^2 + w + 1 = 0$ very easily, by considering
$w^3 -1.$
\item[(c)] If
$z = a + bw$ with $a,b$ real, let $M(z) = (a + bw)(a + bw^2).$
Show that $M(z)$ is real and that for all $z_1, z_2$ we have
$M(z_1)M(z_2).$
\item[(d)] Show that the triangle with sides $3,5,7$ contains an angle
of $2\pi/3.$
\item[(e)] Show that there are infinitely many pair-wise non-similar
triangles with integral sides, and containing an angle of
$2\pi/3.$
\item[(f)] Suppose that $\alpha$ is an angle occuring
in a triangle with integral sides. Is it true that there are
infinitely many pair-wise non-similar
triangles with integral sides, and containing the angle
$\alpha?$
\end{enumerate}
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