Could $2^{1/n}$ be straightedge-and-compass constructible for some $n \in \mathbb{N}$ that is NOT a power of 2?

Question

I understand that the square root of an integer is constructible using straightedge and compass; but are they the only constructible radicals?

Any hint would be greatly appreciated.

Answer 1

No. If $a$ is constructible, then there is a tower of fields $\mathbb{Q} = K_0 \subseteq K_1 \subseteq ... \subseteq K_n \ni a$, with $[K_i : K_{i-1}] = 2$. Therefore, since $[K_n : \mathbb{Q}] = [K_n : \mathbb{Q}(a)] [\mathbb{Q}(a) : \mathbb{Q}]$, $[\mathbb{Q}(a) : \mathbb{Q}]$ is a power of 2. $2^{1/n}$ has minimal polynomial $x^n-2$, and if $n$ is not a power of 2, then $[\mathbb{Q}(a) : \mathbb{Q}]$ is not a power of 2.