Function fields and dominant pullbacks on general varieties

One may describe a rational function as a regular function on a nonempty open, with two representatives identified when they agree on a nonempty open of their common domain. In an irreducible space every finite intersection of nonempty opens is nonempty, so this is an equivalence relation. On an irreducible affine chart $V$, a regular function is locally a quotient of polynomial functions. Any one nonempty such neighborhood therefore represents an element of $\mathrm{Frac}k[V]$, and conversely each fraction is regular on its nonempty denominator open. Equality on a nonempty open implies equality of fractions because the coordinate ring is a domain. This also agrees with the supplied principal-open field identification.

Any two nonempty affine charts meet. Restricting their rational representatives to their overlap identifies both fields with the rational functions just described. This identification is independent of every further restriction, and triple overlaps give the cocycle identity. Affine charts exist by the finite-cover result.

If $f$ is dominant, the inverse image of a nonempty target open is nonempty. For nonempty source open $V$ and target open $W$, $V\cap f^{-1}(W)$ is nonempty whenever $W$ is nonempty; hence a restriction from any nonempty source open remains dominant. Choose affine charts on source and target with the source chart in the target-chart inverse image. The affine dominant pullback is injective. Compatibility with restrictions identifies it with a unique injection between the fields in the statement.