Let $K/F$ be an algebraic field extension. We say that elements $\alpha$ , $\beta$ $\in$ K are conjugate over F if $\alpha$ and $\beta$ have the same minimal polynomial over F.
While reading Field Theory by Pete L.Clark, I came across that if $\alpha$ $\in$ $K$ \ $F$ then the number of conjugates of $\alpha $ over $F$ is at least 2.
My attempt : Suppose $F$ has characteristic $p>0$ .Suppose $\alpha$ $\in$ $K$ such that ${\alpha}^p = a$ for some a $\in F$ . Clearly minimal polynomial of $\alpha$ over $F$ divides f(t) = ${t}^p - a$ $\in F[t]$ . But f(t) = ${(t-{\alpha})}^p \in K[t].$ Hence clearly number of conjugates of $\alpha$ is $1$.
Can anyone explain me where I am going wrong?Any help would be welcome.