Why is it that if $\alpha$ is a root with minimal polynomial of degree $2^k$ (for some integer $k$), then $\alpha$ is constructible?

Question

Why would it be true that if the minimal polynomial of $\alpha$ has degree equal to a power of $2$ then $\alpha$ is constructible? I came across this fact recently while studying Galois theory on my own, but there was no justification given.

Answer 1

It is not true for degrees higher than $2^1=2$. For example, the irreducible equation $x^4-x-1=0$ has no constructible roots. If you try to solve the equation with radicals you get cube roots which you can't simplify, and you're doomed.

For irreducible quartic equations over the integers the roots can be constructed only if the resolvent cubic has constructible roots, which means the resolvent cubic has to have a rational root. This is not true of $x^4-x-1=0$ or in general. If the equivalence breaks down here, there is no justification for assuming it could repair itself for higher power-of-$2$ degrees.