Is $[\mathbb{Q}(x, y):\mathbb{Q}] = 2^r$ a sufficient condition for a point $(x, y) \in \mathbb{R}^2$ to be constructible by ruler and compass?
Let $F$ be a Galois extension over $\mathbb{Q}$ with $[F:\mathbb{Q}]=2^n$, then all elements in $F$ are constructible