[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.