\documentclass{article} \usepackage{latexsym} \newtheorem{definition}{Definition} \newtheorem{lemma}{Lemma} \newtheorem{proposition}{Proposition} \author{Geoff Smith} \title{Extra questions for Chapter 3} \renewcommand{\baselinestretch}{1} \begin{document} \maketitle \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} \end{document}