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.