If there a canonical representation of finite fields $\Bbb F_{p^n}$ for $n>1$?
By canonical I mean that if I were to say to someone else "this bunch of bits represents an element of $\Bbb F_{p^n}$", he'd be able to find that element without needing more information.
In the case $n=1$, it's obvious that the bunch of bits simply represents an integer $a$ less than $p$. And the element is $a+(p)$ in $\Bbb Z / p\Bbb Z$.
But for $n>1$, I didn't see any canonical representation and since we may compute the splitting fields differently, we could end up with isomorphic but different fields. And then the binary representation wouldn't represent the correct element.
Obviously, you could add some order on the polynomials and use that to make sure you get the same representation but that seems rather arbitrary. Hence my question.
The finite field $F_{p^n}$ is a splitting field of the polynomial $f(x) =x^{p^n}-x$ over $\mathbb{Z}_p$. That determines $F_{p^n}$ up to a field isomorphism.
It's ok that there are $F_{p^n}$ that are set-theoretically different but isomorphic, just like there are different ways of defining $\mathbb{R}$. Is "Dedekind cut" canonical or "Completion of $\mathbb{Q}$" canonical or what have you? What you call "canonical representation" is a choice.