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I've recently learned about the field of constructible numbers (those which can be constructed with compass and straight-edge). A theorem in this subject states that a number $z$ (real or complex) is constructible iff the field $\mathbb Q(z)$ is obtainable by a chain of extensions of degree 2.

I wondered, then, if anybody had considered the same situation, but allowing chains of degree $3$, or $n$ for that matter, and then went in reverse to see what "tool" this would correspond to. In other words, if taking degree 2 extensions corresponds to using a straight edge and compass, what new drawing instruments are you introducing by allowing degree 3 extensions, for example?

I've looked online and it seems there's not much talk on the subject. Maybe someone has some insight into this? I might add that it's possible the answer isn't going to be a "tool" that exists, per se (for example it might be "an instrument that can trisect angles", for lack of a better pseudo-example). I'd also add that we'd be looking for the equivalent tool, not just a sufficient one like, "a tool that can draw any algebraic number" (but also something somewhat "geometric", as opposed to "a tool that can find cubic roots", although I'm aware that maybe this is all we can say).

Thanks!

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  • $\begingroup$ A number is in a tower obtained by extensions of degree $2$ and/or $3$ iff it is constructible by marked ruler iff it is constructible by a specific type of origami construction. $\endgroup$ – André Nicolas Dec 1 '14 at 19:07
  • $\begingroup$ @AndréNicolas thanks! Could you elaborate on "origami construction" though? $\endgroup$ – GPerez Dec 1 '14 at 19:10
  • $\begingroup$ There are many definitions of origami construction, not all equivalent. I am sure there is a Wikipedia article. $\endgroup$ – André Nicolas Dec 1 '14 at 19:14
  • $\begingroup$ @AndréNicolas Ah, yes. I thought I would just get arts and crafts pages if I looked. $\endgroup$ – GPerez Dec 1 '14 at 19:21
  • $\begingroup$ There is a moderately well-developed mathematical theory, more precisely theories, with various axiomatizations. $\endgroup$ – André Nicolas Dec 1 '14 at 21:17
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Took a while to find, see https://oeis.org/A005109 With an angle trisector, we can now draw regular $n$-gons when $n$ is a Pierpont prime. The article I could not recall was Gleason,

Gleason, p. 191: a regular polygon of n sides can be constructed by ruler, compass and angle-trisector iff n = 2^r * 3^s * p_1 * p_2 .... p_k, where p_1, p_2,....,p_k are distinct elements of this sequence and >3.

For what it may be worth, I quite like Martin, but I have a different book of his; anyway George E. Martin: Geometric Constructions. Springer, 1998. ISBN 0-387-98276-0.

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