I can't understand the proof of a corollary of the Noether normalisation lemma in Undergraduate algebraic geometry by Miles Reid.
Noether normalisation lemma Let $k$ be an infinite field, and $A=k[x_1,\cdots,x_n]$ a finitely generated $k$-algebra. Then there exists $m\leq n$ and $y_1,\cdots,y_m \in A$ such that
- $y_1,\cdots,y_m$ are algebraically independent over $k$.
- $A$ is finitely generated as $k[y_1,\dots,y_m]$-module.
Corollary Under the hypotheses of the Noether normalisation lemma, there exist $y_1,\cdots,y_m,y_{m+1} \in A$ such that $y_1,\cdots,y_m$ satisfy the conclusion of the lemma, and in addition, the field of fractions $K$ of $A$ is generated over $k$ by $y_1,\cdots,y_m,y_{m+1}$.
In the proof of the corollary, it is first proven that $y_1,\cdots,y_m$ can be chosen so that $K$ is a finite separable extension of $k(y_1,\cdots,y_m)$. And there exists $y_{m+1} \in K$ such that $K= k(y_1,\cdots,y_m,y_{m+1})$. I can understand this part. Then,
$y_{m+1}$ can be taken as a linear combination of $x_i$ with coefficients in $k[y_1,\cdots,y_m]$
Why is that ?