Further Issues
Once KX is defined, it is possible to study properties of X which depend only on KX. This is the subject of birational geometry.
If X is an algebraic variety over a field k, then over each open set U we have a field extension KX(U) of k. The dimension of U will be equal to the transcendence degree of this field extension. All finite transcendence degree field extensions of k correspond to the rational function field of some variety.
In the particular case of an algebraic curve C, that is, dimension 1, it follows that any two non-constant functions F and G on C satisfy a polynomial equation P(F,G) = 0.
Read more about this topic: Function Field (scheme Theory)
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