Fields and new operations

Main Question or Discussion Point

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.
 
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Answers and Replies

AKG
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Technically speaking, a set is not a field, a set together with an addition and a multiplication, (F, +, *) is a field. Since the positive integers, non-negative integers, integers, and rationals are all countable (since there is a bijection between any pair of those sets), and since the rationals are a field, yes, you could make a field using any one of those sets, you'd essentially just be relabelling the rationals in such a way that makes addition and multiplication look very strange. Really, if you could name infinitely many animals, then you can make a field of animals, but what the heck does "elephant + zebra = giraffe" mean?
 

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