Definition of *pure* transcendental extension

Solution 1:

An extension $L/K$ is purely transcendental if there exists some algebraically independent set $S$ such that $L = K(S)$.

Every extension $L/K$ can be written as $K(S)$ for some set $S$ (e.g. $S = L$) but maybe or maybe not with $S$ being algebraically independent.

For instance, with $\mathbf{Q}(x) / \mathbf{Q}$ I can take $S = \{x, x^2\}$ or $S_2 = \{x\}$. One of these is algebraically independent and the other is not.

Or $\mathbf{Q}(\sqrt{2}, x)/\mathbf{Q}$ I can take $S_1 = \{\sqrt{2}, x\}$ or $S_2 = \{\sqrt{2}, x + \frac37\sqrt{2}, x^2, \frac{1 - x + x^2}{1 - \sqrt{2}}\}$ but never with $S$ being algebraically independent.