The classification theorem for finite fields says that for each $n\geq1$ there's just one field $\Bbb F_{p^n}$ with $p^n$ elements up to isomorphism. It's the field made up with the roots of the polynomial $X^{p^n}-X$ in an algebraic closure of $\Bbb F_p=\Bbb Z/p\Bbb Z$.
Concretely, $\Bbb F_{p^n}$ can be realized as the quotient $\Bbb F_p[X]/(P(X))$ where $P(X)$ is any irreducible polynomial of degree $n$.
So, if you start with ${\Bbb F_p}^n$ and you want to endow it with a field structure you can do it in the following two steps:
* Pick a monic irreducibe polynomial $P(X)\in\Bbb F_p[X]$ (monic is not really essential).
* Write ${\Bbb F_p}^n=\Bbb F_p+\Bbb F_px+\cdots+\Bbb F_px^{n-1}$ and (1) define the usual product for the $x^k$; (2) use $P(x)=0$ to decide what $x^n$ should be.
The theory says that you get the "same thing" whatever irreducible polynomial you choose.