prove that x^6 - x^2 + 2 has no constructible roots
First, i let y=x^2, so the function become y^3 - y + 2
then use the theorem, if polynomial f(x) has integer coefficient and a ration root p/q. (p,q)=1, then p|a0 , q|an.
I got p|2 and q|1
g(2) does not equal to zero.
so. the function does not have a rational root.
How do I go on from here..
For is constructible iff there exists a tower of fields in where that and with . Therefore, for some (because ). Let have degree over (meaning the minimal polynomial has degree ). We see that and so for some . Thus, minimal polynomials of constructible numbers must be power of twos.