We can consider nim as a countably infinite-dimensional vector space over , the field with two elements. To do this, we write out a nim heap in binary, and then we make the digits into an element of by putting the least significant bit in the first entry, the second-least significant bit in the second entry, and so forth. In this case, the operation of nim sum is simply vector space addition on .
But we can do even better: we can turn this vector space into a field of characteristic 2. We define multiplication as follows (I will use for nim product): for any , , and a nim product of distinct Fermat 2-powers (numbers of the form ) is their ordinary product. Then we can use distributivity to compute any nim product. For example, . (Note that, unlike nim addition, nim multiplication does not preserve parity.)
In fact, we can say even more. Suppose we limit our game of nim game to piles of size at most . Then nim addition and multiplication are closed in this subclass of games. Therefore we have canonically(?) embedded in our nim field. (I am not so sure about the canonical part, because I don’t know if we could define multiplication in some other natural way that would still give us a field.) This tells us that is isomorphic to the field of finite nimbers.
So here’s a question for myself or my (possibly nonexistent) audience: what does nim multiplication tell us about actually playing games? Anything? Nothing at all? I would like to believe that it has some implication as far as playing games goes, but at the moment, I don’t see anything.
If we extend nim to ordinal piles, we can do something similar and end up with an algebraically closed Field (Conwayese for something that would be a field except that its elements form a proper class). Ordinal nim piles make sense as far as game play goes because the ordinals are (very conveniently!) well-ordered, so games cannot last forever.