why do rational numbers require an extension
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One of the key things you do with algebraic number fields is study them as extensions of Q. (Q itself is trivially an extension of Q.) For example, you can check whether unique factorization exists or not, or represent the algebraic number field as an integral trace form.
R and C are not very well-behaved from this perspective because they can only be represented as infinite-dimensional extensions of Q. It would not be possible to write an integral basis for R or C, so any results that depend on this would not apply.
You could imagine that the definitions were changed a little bit so that R and C and possibly other fields were also called "algebraic number fields" and we distinguished "finite-dimensional algebraic number fields" instead, but the latter appear to be more interesting and fruitful
R and C are not very well-behaved from this perspective because they can only be represented as infinite-dimensional extensions of Q. It would not be possible to write an integral basis for R or C, so any results that depend on this would not apply.
You could imagine that the definitions were changed a little bit so that R and C and possibly other fields were also called "algebraic number fields" and we distinguished "finite-dimensional algebraic number fields" instead, but the latter appear to be more interesting and fruitful
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