Purely transcendental field extension

Definition
Let $$k$$ be a field and $$K$$ be a field extension of $$k$$ (i.e. a field containing $$k$$). Then, we say that $$K$$ is a purely transcendental field extension of $$k$$, if there exists a subset $$T$$ of $$K$$ such that $$T$$ is algebraically independent over $$k$$, and the naturally induced map from the field of fractions $$k(T)$$ to $$K$$ is an isomorphism.