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why is the sin(2*pi/7) non-constructible?
The sine of 2π/7 is classified as a non-constructible number because it does not reside within a quadratic extension of the real numbers (R). A number is deemed constructible if it can be derived using a straight edge and compass, specifically lying in an extension of degree 2n for some integer n. The minimal polynomial for sin(2π/7) is identified as the cyclotomic polynomial x5 + x4 + x3 + x2 + x + 1, which indicates that it cannot be expressed through a sequence of quadratic extensions, thereby confirming its non-constructibility.
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