I am trying to show that $\alpha$ is not an element of $F(\beta)$:
My attempt was by contradiction suppose $\alpha \in F(\beta)$
Then $\alpha = h(\beta) $ where $h$ is a polynomial in $F[x]$
Then $h(\beta)^p=t$
However, I'm stuck at this point.
I need the above to show that $x^p -t$ is irreducible in $F(\beta)$