Corollary

Let A be an integral domain, K its field of fractions and x an element of a K-algebra K (not necessarily commutative). *If x is integral over A, the coefficients of the minimal polynomial $P$ of $x$ over $K$ are integral over $A$ (and they therefore belong to $A$ if $A$ is integrally closed).

There exists by hypothesis (no. 1, Theorem 1) a monic polynomial $Q \in A[X]$ such that $Q(x) = 0$. As $P$ divides $Q$ in $K[X]$, it follows from Proposition 11 that the coefficients of $P$ are integral over $A$.

Let $A$ be a ring and $R$ a (commutative) $A$-algebra; the homomorphism $\phi : A \to R$ defining the $A$-algebra structure on $R$ can be extended uniquely to a homomorphism $A[X] \to R[X]$ of polynomial rings over $A$ and $R$, leaving $X$ invariant and hence $R[X]$ is given a canonical $A[X]$-algebra structure.