Let $F_p$ be a finite field having $p$ elements(for some prime $p$), not having primitive $3$rd root of unit say $\omega.$ Then can i say that field $F_p(\omega)$ and $F_{p^2}$ are isomorphic.
I tried as follows: As both of fields $F_{p^{2}}$ and $F_p(\omega)$ are of same dimension $2$ over the field $F_p$ and hence same number of elements. Now any two finite field with same number of elements are isomorphic.
Please suggest me. Thanks in advance.