Chapter Questions
If $a$ and $b$ are constructible numbers and $a \geq b>0$, give a geometric proof that $a+b$ and $a-b$ are constructible.
If $a$ and $b$ are constructible, give a geometric proof that $a b$ is constructible. (Hint: Consider the following figure. Notice that all segments in the figure can be made with an unmarked straightedge and a compass.)
Prove that if $c$ is a constructible number, then so is $\sqrt{|c|}$. (Hint: Consider the following semicircle with diameter $1+|c| .$ ) (This exercise is referred to in Chapter $33 .$.)
If $a$ and $b(b \neq 0)$ are constructible numbers, give a geometric proof that $a / b$ is constructible. (Hint: Consider the following figure.)
Prove that $\sin \theta$ is constructible if and only if $\cos \theta$ is constructible.
Prove that an angle $\theta$ is constructible if and only if $\sin \theta$ is constructible.
Prove that $\cos 2 \theta$ is constructible if and only if $\cos \theta$ is constructible.
Prove that $30^{\circ}$ is a constructible angle.
Prove that a $45^{\circ}$ angle can be trisected with an unmarked straightedge and a compass.
Prove that a $40^{\circ}$ angle is not constructible.
Show that the point of intersection of two lines in the plane of a field $F$ lies in the plane of $F$.
Show that the points of intersection of a circle in the plane of a field $F$ and a line in the plane of $F$ are points in the plane of $F$ or in the plane of $F(\sqrt{\alpha})$, where $\alpha \in F$ and $\alpha$ is positive. Give an example of a circle and a line in the plane of $Q$ whose points of intersection are not in the plane of $Q$.
Prove that $8 x^{3}-6 x-1$ is irreducible over $Q$.
Use the fact that $8 \cos ^{3}(2 \pi / 7)+4 \cos ^{2}(2 \pi / 7)-4 \cos (2 \pi / 7)-1=0$ to prove that a regular seven-sided polygon is not constructible with an unmarked straightedge and a compass.
Show that a regular 9 -gon cannot be constructed with an unmarked straightedge and a compass.
Show that if a regular $n$ -gon is constructible, then so is a regular $2 n$ -gon.
(Squaring the Circle) Show that it is impossible to construct, with an unmarked straightedge and a compass, a square whose area equals that of a circle of radius 1 . You may use the fact that $\pi$ is transcendental over $Q$.
Use the fact that $4 \cos ^{2}(2 \pi / 5)+2 \cos (2 \pi / 5)-1=0$ to prove that a regular pentagon is constructible.
Can the cube be "tripled"?
Can the cube be "quadrupled"?
Can the circle be "cubed"?
If $a, b$, and $c$ are constructible, show that the real roots of $a x^{2}+$ $b x+c$ are constructible.