On subfields of rational function fields (Q798370)

Let \(k\subset K\) and \(k\subset L\) be field extensions. Suppose that \(L\) is contained in a purely transcendental extension of \(K\). Then \(L\) is \(k\)-isomorphic to a subfield of \(K\), provided the degree of transcendency of \(L| k\) satisfies: \(dt L| k\leq dt K| k.\) This statement constitutes as useful lemma in certain investigations of unirational fields. Until now, however, it was proved in the literature for infinite base field \(k\) only. The proof presented here is valid also for finite base field.