I'm trying to solve the following:
Let $K$ be the smallest field, with characteristic $2$ such that it contains a $15$-primitive root of the unit. Find its cardinality and a primitive element of such a field.
I know that a $15-$th root of unity is contained in such a field if $$15|q-1$$ where $q$ is the cardinality of the field. Now I should use the condition $\chi(K)=2$, but I really don't know how. I mean, how can use it?
I would be tempted to say $q = 2^n$, but does this really imply that the characteristic is $2$? I just know $F_2$ as field with such a char, it seems wrong to me actually.
Once I get what is the field, it's just a matter of computation to find the primitive element.