f_{12} = (x - ζ)(x - ζ^{5})(x-ζ^{7})(x-ζ^{11})is the definition. However the surprising thing about cyclotomic polynomials is that the coefficients are all rationals (integers infact), none of the surds from taking twelfth roots escape to pollute the polynominal. My book proves this by giving an inductive construction.
Start from x^{12}-1 and divide by all of f_{1}, f_{2}, f_{3}, f_{4}, f_{6}, the point being that 1,2,3,4, and 6 are divisors of 12.
Now I've always believed that ζ has a twelfth degree minimum polynomial and that the field extension [Q(ζ):Q] is twelfth degree. It is "just obvious", I never really thought about it, and I don't spot what is coming.
Starting as I said from x^{12}-1 I divide by x-1 getting
x^{11}+ x^{10}+ x^{9}+ x^{8}+ x^{7}+ x^{6}+ x^{5}+ x^{4}+ x^{3}+ x^{2}+ x+ 1Next comes division by x+1 leaving
x^{10}+ x^{8}+ x^{6}+ x^{4}+ x^{2}+ 1Dividing by x^{2}+x+1 requires care, so I get
x^{8}- x^{7}+ x^{6}+ x^{2}- x+ 1without seeing the danger. Dividing by (x^{2}+1) is an easier calculation but I'm feeling smug that things are going well so I'm not smelling the burning insulation yet. Finally division,by x^{2}-x+1 reveals
x^{4}- x^{2}+1
Oh shit! How the fuck did that happen. ζ is supposed to be a twelfth root, what is it doing satisfying a fourth degree polynomial?
Starting to panic I fire up my trusty REPL and start coding up numerical calculations. Soon it is very clear. The twelfth root of unity does satisfy a fourth degree polynomial. [Q(ζ):Q] = 4 not 12. I don't understand mathematics.
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