Why is sin(2*pi/7) a Non-Constructible Number?

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SUMMARY

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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why is the sin(2*pi/7) non-constructible?
 
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because 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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