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 because $X$ is irreducible. By [F4], their overlap contains a nonempty affine principal open in either chart. Restricting rational representatives to that common open and applying [F2] identifies both fields. Compatibility under further restriction makes this identification independent of the chosen common open, 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: the image of $V$ is dense in the irreducible target, so it meets $W$. Choose an affine target chart $W$ and then, inside $f^{-1}(W)$, an affine source chart by [F1] and [F4]. This restriction is dominant, so [F3] gives an injective affine function-field pullback. The restriction compatibility from steps 1.1–2.1 and [F3] identifies it with a unique injection between the general fields. The use of AC is inherited exactly at [F2] and [F3].