Definition 2

Let E be an extension of a field K. A family $(x_i)_{i \in I}$ of elements of E is called a pure basis of E (over K) if it is algebraically free and $E = K(x_i)_{i \in I}$. The extension E of K is also called pure if it possesses a pure basis.

The empty family is algebraically free, hence K is a pure extension of itself. With the notation of Def. 2, every element $x_i$ is transcendental over K; if $I$ is non-empty, E is thus a transcendental extension of K.