I’ll share one I’ve been pondering for 30 years since my geometry teacher told me it’s impossible: It’s not possible to trisect an arbitrary angle using only a straightedge and compass.
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))
(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).