Function field of a variety

Summary

The function field $k(X)$ of an algebraic variety $X$ over $k$ is the field of rational functions on $X$. A rational function is a function regular on some nonempty open subset. Two of them are identified when they agree on a smaller open set. For an affine variety it is the field of fractions of the coordinate ring. It is a finitely generated field extension of $k$. It is a birational invariant: two varieties have isomorphic function fields over $k$ exactly when they are birational.