A field such that no extension has an 11th primitive root of unity I am asked to find the field described in the title. However, I can't quite understand the question. 
For any field $K$, and $\zeta_{11} = e^{\frac{2\pi i}{11}}$ then surely the extension $K \leq K(\zeta_{11})$ contains the 11th primitive root of unity?
So then how can such a field exist?

Hint: what is the order of $\mathbb{F}_q^{\times}$, the roots of unity of a finite field $\mathbb{F}_q$?
