Generalization To Arbitrary Scheme
In the most general setting, that of modern scheme theory, we take the latter point of view above as a point of departure. Namely, if X is an integral scheme, then every open affine subset U is an integral domain and, hence, has a field of fractions. Furthermore, it can be verified that these are all the same, and are all equal to the local ring of the generic point of X. Thus the function field of X is just the local ring of its generic point. This point of view is developed further in function field (scheme theory).
Read more about this topic: Function Field Of An Algebraic Variety
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