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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).



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.




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