Let be the quotien field of
Exhibit a transcendence base for over , and express explicitly as an algebraic extension over .
let and the claim is that is a transcendence basis for over : since the element is
algebraic over and thus is algebraic over so to prove that is a trancendence basis for over we only need to show that is algebraically independent over so suppose that
the elements of are algebraically dependent over then there exists such that i.e. which is obviously impossible because
every non-zero element of involves but does not.