Given any variety $X$, we can get the field of rational functions $\mathbb{C}(X)$. If we take any non-empty open subset of $U$, we get another field of rational fractions $\mathbb{C}(U)$. These fields seem to be isomorphic (see Cox Little Schenck exercise 3.0.4), which makes sense to me, but I don't know how to prove it.
I found a similar question here, but this solves the above question only for affine open subsets of $X$. Do you have any suggestions, how this can be solved for arbitrary open subsets $U\subset X$?
The field of rational functions is the stalk of the structure sheaf at the generic point. The generic point of $U$ is the same as the generic point of $X$, so the stalk at that point is the same.