Proposition 12

Let $A$ be a ring, $R$ an $A$-algebra and $P$ a polynomial in $R[X_1, \ldots, X_n]$. For $P$ to be integral over $A[X_1, \ldots, X_n]$, it is necessary and sufficient that the coefficients of $P$ be integral over $A$.

By considering the polynomials of $R[X_1, \ldots, X_n]$ as polynomials in $X$, with coefficients in $R[X_1, \ldots, X_{n-1}]$, we see immediately that it is reduced to proving the proposition for $n = 1$. Then let $P$ be a polynomial in $R[X]$; it follows immediately from no. 1, Proposition 5 that, if the coefficients of $P$ are in the integral closure $B$ of $A$ in $R$, the element $P$, which belongs to $B[X] = B \otimes_A A[X]$, is integral over $A[X]$. Conversely, suppose that $P$ is integral over $A[X]$ and let

be a monic polynomial with coefficients $F_i \in A[X]$ with $P$ as a root. Let $r$ be an integer strictly greater than all the degrees of the polynomials $P$ and $F_i$ ($1 \leq i \leq m$) and let us write

Then $P_1$ is a root of the polynomial

with coefficients in $A[X]$; we may therefore write

Now the choice of $r$ implies that $-P_1$ is a monic polynomial of $R[X]$ and so is $G_m(X) = Q(X^r)$, the degrees of the polynomials $F_k(X) X^{r(m-k)}$ being all $< rm$ for $k \geq 1$. We conclude first of all that the polynomial

of $R[X]$ is also monic; moreover, as the coefficients of $G_i$ belong to $A$, Proposition 11 shows that $P_1$ has coefficients integral over $A$ and the coefficients of $P$ are therefore certainly integral over $A$.