Alternate descriptions of finite fields The finite field of order $p^n$ is isomorphic to $(\mathbb Z/p \mathbb Z)[X]/(P)$, where $P$ is an irreducible polynomial in $(\mathbb Z/p \mathbb Z)[X]$ of degree $n$. This describes every finite field up to isomorphism.
My question is, what are alternate ways of describing these finite fields?
For example, Conway gave an alternate way of describing the fields $GF(2^{2^n})$ (see this example answer).
 A: In ONAG chapter 6, Conway gives a way of defining the finite field $2^{2^n}$ for any natural $n$. The domain is $\{0,1,\dots, 2^{2^n}-1\}$. Addition is defined by the two rules:


*

*The sum of $2^x$ and $2^y$ with $x \ne y$ is just the regular $2^n+2^m$. (So $13 = 8 + 4 + 1$.)

*The sum of two equal numbers is $0$.


Multiplication is likewise defined by two rules.


*

*The product of $2^{2^x}$ and $2^{2^y}$ with $x \ne y$ is just their ordinary product.

*The square of $2^{2^x}$ is the ordinary product of $\frac 32$ and $2^{2^x}$.


This uniquely defines the field. For example, using only the field axioms and above rules, we conclude $$5 \times 9 = (4+1)(4\times2+1) =\\ 4^2 \times 2 + 4 \times 2 + 4 + 1 = 6 \times 2 + 8 + 4 + 1 =\\ (4 + 2) \times 2 + 13 = 4 \times 2 + 2^2 + 13 =\\8 + 3 + 13 = 6$$
Also see this addition and multiplication table.
Conway goes on to extend this to the set $\{\alpha:\alpha < \beta\}$ for certain ordinals $\beta$, and to the class of all ordinals to create the curious Field $\text{On}_2$.
One nice thing about this construction is it shows show $GF(2^{2^n})$ is isomorphic to a subfield of $GF(2^{2^m})$ for any $n \le m$.
A: It's only a tiny bit different, but if you put $\omega=\exp(2\pi i/(p^n-1))$ and $A=\mathbb{Z}[\omega]<\mathbb{C}$ then $A/pA$ is a product of fields, each of order $p^n$ (and therefore isomorphic to each other).
