91Ó°ÊÓ

Determine whether the indicated real number is constructible. $$ \sqrt{3+\sqrt[4]{5}} $$

In the real Cartesian plane, suppose we are given the fixed parabola \(y=x^{2}\) and, in addition to the other operations in straightedge and compass constructions, we are allowed to mark the intersection of lines and circles that have been drawn with this parabola. Show that then the problem of the duplication of the cube can be solved, and more generally the cube root of any real number is constructible using the parabola.

Given a line segment of length 1 , construct with straightedge and compass a line segment of the indicated length. $$ 2 / 3 $$

Given a line segment of length 1 , construct with straightedge and compass a line segment of the indicated length. $$ \sqrt{1+\sqrt{3}} $$

Call a complex number \(a+b i\) constructible using the marked ruler and compass if its real and imaginary parts \(a\) and \(b\) are both constructible using the marked ruler and compass. Show that if the complex number \(a+b i\) is constructible using the marked ruler and compass, then so are its cube roots.

Show that finding the points of intersection of a line or circle with the parabola \(y=x^{2}\) amounts to solving a quadratic or a quartic equation whose coefficients are rational combinations of the coefficients of the equation of the line or circle in question.

Show that in a regular 5 -gon with sides of length 1 , any diagonal has length $$ \alpha=1 / 2(1+\sqrt{5}) $$

Show that a regular heptagon is not constructible using only a straightedge and compass. (Hint: Show that \(2 \cos ^{2 \pi} / 7\) is a zero of \(x^{3}+x^{2}-2 x-1\).)