Finite nimbers are a way of turning the natural numbers (finite ordinals) into a characteristic 2 field. Addition in this field is found by writing the numbers in binary and adding without carry, while multiplication is harder to describe concisely.
Nimber addition and multiplication have some recursive definitions as follows. Letting mex($S$) the smallest natural number not in $S$, $$ a+b=\text{mex}(\{a'+b\,|a'<a\}\cup\{a+b'|b'<b\}) $$ $$ ab = \text{mex}\,\{ab'+a'b-a'b'|a'<a,b'<b\} $$
I was wondering if anyone knew of a corresponding way to turn the natural numbers into a characteristic 3 field, and perhaps more generally one of characteristic $p$. The reason I'm hopeful this is possible is because there is a similar recursive definition of base 3 addition without carry, namely, $$ a+b = \text{mex}\,(\{a'+b\}\cup\{a+b'\}\cup\{a''+b''|a''+b=a+b''\}) $$ where $a',a''<a,b',b''<b$. There is also a similar definition for all prime $p$. But I can't seem to figure out a definition of multiplication. Has anyone else heard of/thought of this?
You should see the paper "On On_p", by Joseph DiMuro. In it, he gives a characteristic $p$ analogue of the nimbers. The addition is given by adding base $p$ without carries (including for ordinals). The definition of multiplication, unfortunately, is not so simple. But I think it's a good generalization.
The definition for the natural numbers is almost identical to the one Slade gives above; the only difference is in the choice of irreducible polynomial. This will result in a different field structure -- DiMuro's is equal to the union of $\mathbb{F}_{p^{2^k}}$, while (if I'm understanding correctly) Slade's is equal to the union of $\mathbb{F}_{p^{p^k}}$. But it's clearly very similar.