I am having a bit of an issue with an example I was looking at. The question states: "What is the smallest field of characteristic 2 that contains a primitive 11th root of unity?". I am not familiar with how we would go about finding such a field, but my thought processes is going a bit like this:
1) By 'smallest field', I am thinking they mean a field of smallest order? Therefore I want to find the smallest ordered field, with characteristic 2, that contains a primitive 11th root of unity.
2) Having a primitive 11th root of unity means that there $\exists x \in \mathbb{F}_q$ such that $x^{11}=1$ and that $x^s\ne 1$ for $0\lt s \lt 11$. I am not quite sure how I would go about finding the $\mathbb{F}_1$ that contains this primitive root of unity, other than manually checking each $\mathbb{F}_q$, $q=2,3,...$.
3) The fact is given that the characteristic of $\mathbb{F}_q$ is $p=2$. I know that this means that in the field, $1+1=0$, I am not really sure how I would use this fact to find the field. Even so, other than $\mathbb{F}_2$, I am not sure what other fields would have this characteristic.
So yeah, I am a bit lost overall how I would go about attacking this problem (or similar ones - different characteristic, or different n-th roots).
Thanks for any guidance.