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 $n \le m$.