Corollary

— For an extension E of a field K to be pure it is necessary and sufficient for E to be K-isomorphic to a field of rational fractions over K. More precisely, if the family $(x_i)_{i \in I}$ is a pure basis of E, then there exists a unique K-isomorphism of $K(X_i)_{i \in I}$ onto E which maps $X_i$ to $x_i$ for each $i \in I$.