Do the field of complex numbers arise necessarily and uniquely as the only field of pairs of ordered real...
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Let me first assess that I'm not an expert on the subject, so I galdly welcome edits or suggestion, and don't be too mad at me if my assumptions are mistaken. The field of complex number is a set of ordered pair of real numbers equipped with some additional proprieties ( which makes it a field indeed). Now, let's say that we didn't come up with the idea of complex numbers through the study of polynomials. Instead we want to create (for our own fun) a set of ordered pair of real numbers (x,y) with some additional structure/proprieties that makes it behave nicely as our field of real numbers and in addition, it has the property that the subset of all ordered pair (x,0) behave exactly as our beloved field of real numbers under any operation we take. So, we want a field of ordered