the function field

On a variety you would like to divide functions, but a quotient like 1/x is not defined where the denominator vanishes. The function field is what you get when you allow such quotients anyway, accepting that each one is only defined away from a thin bad set. It is the largest field of fractions you can build from the regular functions, and it captures everything about the variety that survives 'up to a lower-dimensional mess' — exactly the birational information.

Let X be an irreducible affine variety, so its coordinate ring k[X] is an integral domain. The function field k(X) is the field of fractions of k[X]: elements are ratios f/g with f, g in k[X] and g not the zero function, two ratios being equal when they cross-multiply to agree. Such an element is a rational function on X; it is regular (genuinely defined) at every point where some representative has nonvanishing denominator, and its domain of definition is a dense open set. For a projective irreducible variety one defines k(X) the same way using any affine chart — the result is independent of the chart. The transcendence degree of k(X) over k equals the dimension of X, which is one of the cleanest definitions of dimension.

The function field is the home of birational geometry: two irreducible varieties are birationally equivalent exactly when their function fields are isomorphic as k-algebras, so k(X) is a complete birational invariant. It does not see the variety's points individually or its singularities locally — those are lost when you allow rational functions — but it sees dimension, the field of rational maps, and whether the variety is rational (k(X) purely transcendental). Caveat: a rational function is not a function on all of X; it has an honest domain, and forgetting the locus where the denominator vanishes leads to false claims, especially when composing rational maps that may not be defined where you want.