These also provide a short proof that cos(20 degrees) + i sin(20 degrees) is not a constructible number (that is, expressible using {0, 1, +, -, *, /, sqrt}): Let the Eisenstein rationals be our base field; observe that the Eisenstein rationals are themselves constructible; observe that cos(20 degrees) + i sin(20 degrees) is Eisenstein-irrational by using unique factorisation; observe that the Eisenstein integer cos(60 degrees) + i sin(60 degrees) has precisely 3 cube roots in the complex numbers; conclude that adjoining cos(20 degrees) + i sin(20 degrees) to the Eisenstein rationals results in a degree 3 extension; a degree 3 extension is not constructible.
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Reminds me of isometric graphics.
These also provide a short proof that cos(20 degrees) + i sin(20 degrees) is not a constructible number (that is, expressible using {0, 1, +, -, *, /, sqrt}): Let the Eisenstein rationals be our base field; observe that the Eisenstein rationals are themselves constructible; observe that cos(20 degrees) + i sin(20 degrees) is Eisenstein-irrational by using unique factorisation; observe that the Eisenstein integer cos(60 degrees) + i sin(60 degrees) has precisely 3 cube roots in the complex numbers; conclude that adjoining cos(20 degrees) + i sin(20 degrees) to the Eisenstein rationals results in a degree 3 extension; a degree 3 extension is not constructible.