I tried f₁₂, the twelfth cyclotomic polynomial. Let ζ be the primitive twelfth root of unity.

f₁₂ = (x - ζ)(x - ζ⁵)(x-ζ⁷)(x-ζ¹¹)

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 x¹²-1 and divide by all of f₁, f₂, f₃, f₄, f₆, 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 x¹²-1 I divide by x-1 getting

x¹¹+ x¹⁰+ x⁹+ x⁸+ x⁷+ x⁶+ x⁵+ x⁴+ x³+ x²+ x+ 1

Next comes division by x+1 leaving

x¹⁰+ x⁸+ x⁶+ x⁴+ x²+ 1

Dividing by x²+x+1 requires care, so I get

x⁸- x⁷+ x⁶+ x²- x+ 1

without seeing the danger. Dividing by (x²+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 x²-x+1 reveals

x⁴- x²+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.