It's well known that angle trisection cannot be done with straightedge and compass alone, as

Theorem 1.If $z \in \mathbb C$ is constructible with straightedge and compass from $\mathbb Q$, then $$\mathbb Q (z) : \mathbb Q = 2^n.$$But the minimal polynomial of $\cos 20 ^{\circ}$ is $8 x ^ { 3 } - 6 x - 1$, so $$\mathbb Q (\cos 20 ^{\circ}) : \mathbb Q = 3,$$

That proves we cannot trisect $ 60 ^{\circ}$.

However, it's doable with origami, as Huzita Axiom 6 - Computing the Origami Trisection of an Angle shows. My question is:

Exactly what field extensions can be obtained by considering origami constructible number? Is this as well-studied as straightedge and compass, i.e. do we have a similar theorem as

Theorem 1?