Will Savin do you know where I can find an article on this "first field"?

Do you mean that $g+g=0$.

Why was my question downvoted? What problem should I address?

Thank you very much for your comment. It is very detailed and it helped me further understand the arguments of Will Sawin and Douglas Zare. I have selected it as the "answer" for further reference to others. It is very extensive and so will give them a full understanding.

Thank you very much for your answer. The only reason this is not the accepted answer is because I accepted before it was posted. I think that the line : "XOR describes the additions with respect to any basis, not just the standard one." is very helpful to anyone (I hope/think I am not the only one) who was confused as to the origin of the nimsum.

There are many binary functions that have similar degrees of minimalism. What specific properties of the xor allow it to be the solution to Nim.

Thank you very much @Will Savin. This is exactly what I was looking for: +1

I see your point Per Alexandersson. However, to find such games, very contrived rules would need to be devised. What qualities then of the xor yield such a natural and minimalist game such as Nim?