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Let $F$ be an infinite field, and $R$ a quotient of $F[x_1,...,x_n]$. Then, there is a subring $S$ of $R$ s.t. $S\cong F[t_1,...,t_r]$ and $R$ integral over $S$.

Proof : We use induction on $n$. If $n=1$, then $R\cong F[x_1]/(f(x_1))$ for some $f(x_1)\in F[x_1]$. By dividing by the leading coefficient of $f(x_1)$, we may assume that it is monic, so we may choose $S=F$.

Q1) Is $f(x_1)$ necessarily irreducible ?

Q2) Why $R$ is integral over $F$ ? Let $q\in R$, i.e. $q=p(x_1)+f(x_1)R[x_1]$. Which monic polynomial will annihilate $q$ ?

Let $n>1$ and suppose the statement is true for $m<n$. Let $\bar x_i$ the residue classes of $x_i$ in $R$. We may assume after reordering the variables that $\bar x_n$ is algebraic over $F(\bar x_1,...,\bar x_{n-1})$. Indeed, otherwise, $\bar x_i$ are algebraically independent, and hence we may choose $S=F$.

Q3) Why $\bar x_i$ are algebraically independent, and hence we may choose $S=F$ ?

So there is a polynomial $g(y_1,...,y_n)\in F[y_1,...,y_n]$ s.t. $g(\bar x_1,...,\bar x_n)=0$ and $g(\bar x_1,...,\bar x_{n-1},y_n)\neq 0$ as a polynomial in $y_n$. Let $G$ be the highest degree homogeneous par of $g$, and let $d$ be it's degree.

Q4) I don't understand what mean "Let $G$ be the highest degree homogeneous par of $g$, and let $d$ be it's degree". What is $G$ ?

If I know what is $G$, I think that the rest of the proof will be ok.

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  • $\begingroup$ I can't see any finite field in your question. $\endgroup$ – user26857 Dec 25 '16 at 20:20

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