How does one endow $\mathbb GF(p)^{n}$ with a field structure?

Question

I've tried to endow $\mathbb{F}^{n}$, where $\mathbb{F}=\mathbb GF(p)$ with a field structure, but I was not able to do it. Could you please help me with this question?

Answer 1

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.