Non-constructible number


by keiop3
Tags: nonconstructible, number
keiop3
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#1
Apr28-04, 02:04 AM
P: 1
why is the sin(2*pi/7) non-constructible?
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matt grime
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#2
Apr28-04, 03:53 AM
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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
mathwonk
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#3
Sep16-04, 11:30 PM
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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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