Discussion
Which of the following constructions is not possible by ruler and compass ?
Answer: B
The construction that is not possible is a regular polygon of 7 sides.
By Gauss and Wantzel, a regular \(n\)-gon is constructible with ruler and compass exactly when \(n\) is a power of 2 times a product of distinct Fermat primes (3, 5, 17, 257, 65537). 7 is prime but not a Fermat prime; equivalently \(\cos(2\pi/7)\) has degree 3 over Q, not a power of 2.
The others are possible. Trisecting 90° means constructing 30°, which is easy. \(1 + \sqrt{2} + \sqrt[4]{2}\) uses only repeated square roots. And \(e^{\pi i/40}\) is a vertex of the regular 80-gon, and \(80 = 2^{4} \times 5\) with 5 a Fermat prime, so it is constructible.
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