Transcendence bases are a useful tool to prove various existence statements about field homomorphisms. Here is an example: Given an algebraically closed field "L", a subfield "K" and a field automorphism "f" of "K", there exists a field automorphism of "L" which extends "f" (i.e. whose restriction to "K" is "f"). For the proof, one starts with a transcendence basis "S" of "L"/"K". The elements of "K"("S") are just quotients of polynomials in elements of "S" with coefficients in "K"; therefore the automorphism "f" can be extended to one of "K"("S") by sending every element of "S" to itself. The field "L" is the algebraic closure of "K"("S") and algebraic closures are unique up to isomorphism; this means that the automorphism can be further extended from "K"("S") to "L".