why is the sin(2*pi/7) non-constructible?
Becuase it doesn't lie in a quadratic extension of a quadratic extension of (etc) R: a number is constructible iff (using straight edge and compass) if it lies in an extension of degree 2^n for some n. The proof is elementary and a good exposition can be found in almost any Galois THeory book. To check this particular example find the minimal polynomial of sin2pi/7, which i imagine is the cycltomic x^5+x^4+x^3+x^2+x+1
constructible means it is obtained by intersecting some lines and circles, hence given by quadratic equations. thus a sequence of extension fields of degree 2. since field extension degree is multiplicative, repeating them gives fields of degree 2^n. so any number satisfying an irreducible equation of degree not a power of 2 is not constructible.
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