Exercise 3, § 3

Let K be an infinite field, L an extension field of K, $(x_i)_{1 \leq i \leq n}$ a finite family of elements of L and $d$ the transcendence degree of $K(x_1, \ldots, x_n)$ over K. Show that there exist $d$ elements $z_j = \sum_{i=1}^n a_{ji} x_i$ of L (where the $a_{ji} \in K$) such that $K[x_1, \ldots, x_n]$ is integral over $K[z_1, \ldots, z_d]$. If we suppose further that $K(x_1, \ldots, x_n)$ is a separable extension of K, show that the $a_{ji}$ may be chosen such that the $z_j$ form a separating transcendence basis of $K(x_1, \ldots, x_n)$ over K (argue by induction on $n$ when $n > d$).