Let $F$ be a field of characteristic 2. I need to show the existence of a quadratic polynomial in $F[t]$ which cannot be solved by adjoining all square roots of elements in the field.
Attempt:
For $F=\mathbb{Z}_2$, $f(t)=t^2+t+1$ works since $\mathbb{Z}_2$ is closed taking square roots.
I don't know how to do the general case. I think that the same polynomial could work.
Edit: Robert shows a counterexample in comments.
If $F$ is algebraically closed, then there exists no such $f$, since any $f$ splits. The polynomial $f(t)=t^2+t+1$ works for $\mathbb{Z}_2$, but not in general (take $F$ to be the splitting field of this polynomial). Maybe there are other conditions you can impose on $F$ so that this is true?