Algebraically independent elements

Elements $x_1,\dots ,x_n$ of a field extension of $K$ are algebraically independent over $K$ when no nonzero polynomial in $K[X_1,\dots ,X_n]$ vanishes at $(x_1,\dots ,x_n)$.

The rational function field $K(x_1,\dots ,x_n)$ is the field of fractions of $K[x_1,\dots ,x_n]$ when the variables are algebraically independent over $K$.