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\section*{Math 403/503 Spring 2024}
\subsection*{Homework 10, due April 24}
\begin{enumerate}
\item Show that if $\alpha \in \R$ is a positive and constructible number, then $\sqrt{\alpha}$ is also constructible. (In other words, $\R_{>0}\cap\R_\text{c}$ is closed under square roots.)
\medskip
\item Show that the regular $9$-gon is not constructible with straightedge and compass (without actually performing the construction).
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\item Can a cube be constructed with three times the volume of a given cube?
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\item Show that if $A=(a_1,a_2), B=(b_1,b_2), C=(c_1,c_2), D=(d_1,d_2)$ are four distinct points in the plane, such that the lines $\overline{AB}$ and $\overline{CD}$ intersect at $Q=(q_1,q_2)$, show that $q_1$ and $q_2$ belong to the field $\Q(a_1,a_2,b_1,b_2,c_1,c_2,d_1,d_2)$.
\medskip
\item Prove that the cosine of one degree ($\cos 1^\circ$) is algebraic over $\Q$ but not constructible.
\end{enumerate}
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