[redvenom]

The reason basically comes down to the fact that every time you perform a compass and straight-edge construction, you are constructing quadratic extensions of number fields, whereas trisection (in general) needs a cubic extension.

The 'in general' part is important: of course, you can trisect specific angles like a 90 degree angle.

(The construction of number field extensions is happening because let's say you construct a 90 degree angle, well that allows you to construct an isosceles triangle with equal sides length 1, and therefore you construct the square root of 2, so the extension Q(sqrt(2))

[murkle]

You need to be careful how you phrase it though - it's possible with a _marked_ edge https://www.geogebra.org/m/cWfHr7pk

[dmurray]

That's a long way from being a convincing argument that compass and straight-edge can only produce quadratic extensions to number fields, though.

(I know it's true, just saying it looks like you've offered an explanation in elementary terms, but I don't think one exists).

[redvenom]

Absolutely true, there is a lot to prove there. It's not an elementary explanation by any means, rather it is an outline that someone who knows Galois theory well could probably reconstruct.