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I use the word «field» in purely algebraic sense here. Sometimes, when reading textbooks I encounter sentences like «Although the formulae in this section derived for the field of real numbers, they remain valid for complex numbers field as well». Or even more general variant of it: «...remain valid for any field» (of course, I am not talking here about standard operations like addition, multiplication and so on, which make it possible to for a set to be called a field). Frankly, I can't give you an example of such textbooks right now, but I feel quite sure that I encountered such phrases not once (just did not try to ponder them over than). What does it mean for a relation «to be valid for any field»?
And the opposite question in some sense: obviously, different fields have much in common, but they also possesses some «fine internal structure» that makes them qualitatively different from each other. Is it possible to formalise and understand those structure differences? OK, the complex numbers field does behave differently from its real numbers counterpart, but it is just a statement, a fact, but why it is so, can the differences be predicted before studying behaviour of both fields? For example, can it be, that differences between real and complex analyses stem from the fact one of these fields is ordered while the other is not?
And the opposite question in some sense: obviously, different fields have much in common, but they also possesses some «fine internal structure» that makes them qualitatively different from each other. Is it possible to formalise and understand those structure differences? OK, the complex numbers field does behave differently from its real numbers counterpart, but it is just a statement, a fact, but why it is so, can the differences be predicted before studying behaviour of both fields? For example, can it be, that differences between real and complex analyses stem from the fact one of these fields is ordered while the other is not?