This problem comes from Halmos's Finite Dimensional Vector Spaces. Given that we can re-define addition or multiplication or both, is the set of all nonnegative integers a field? What about the integers? My thinking is that since the Rational numbers form a field, and they are countable, we can assigen each number of the aforementioned set a rational and then we can have a field with the integers representing rationals. Am I wrong? Edit: On second thought, this doesn't seem to right. Uniqueness is violated somewhere i think.