Ring of polynomials = $\mathbb{Q}[x]$
I'm having a confusion with the theory of minimum polynomials.
Say is $z$ is an $11^{th}$ root of unity. (That is: $z^{11}=1$).
And I know because of the cyclotomic polynomial, we have $z^{11}-1=(z-1)(z^{10}+z^{9}+...z+1)$.
1. Hence it is not irreducible in $\mathbb{Q}[x]$ Am I correct?
2. Hence we can say that the minimum polynomial can be of degree less than $11$. Is that correct?
3. And is there any more detail that I can obtain about the minimum polynomial of $z$?
4.And also how about the $[\mathbb{Q}[z]:\mathbb{Q}]$ (That is the dimension of the vector space adjoining $z$ to $\mathbb{Q}$)
5. Also it always gives me the doubt why z can not be equal to one... is it because the roots of unity are defined not to be equal to one?
I really appreciate your help