Proposition 11

Let A be a ring, R an A-algebra and P and Q monic polynomials in R[X]. If the coefficients of PQ are integral over A, the coefficients of P and Q are integral over A.

By a double application of Lemma 2, we see that there exists a ring R' containing R and families of elements $(a_i)_{1 \leq i \leq m}, (b_j)_{1 \leq j \leq n}$ of R' such that in $R'[X]$

the coefficients of P Q belong to the integral closure $A'$ of A in R' and hence (no. 2, Proposition 7) the elements $a, (1 \leq i \leq m)$ and $b, (1 \leq j \leq n)$ belong to A. It follows that the coefficientsof P and Qare integral over A (no. 1, Corollary 1 to Proposition 4).

Let A be an integral domain, K its field of fractions and K' a K-algebra (not necessarily commutative). Given an element $x \in K'$ algebraic over K, the polynomials $P \in K[X]$ such that $P(x) = 0$ form an ideal $a \neq 0$ of $K[X]$, necessarily principal (Algebra, Chapter IV, § 1, no. 5, Proposition 7). There exists a unique monic polynomial which generates a; generalizing the terminology introduced in Algebra, Chapter V, § 3, no. 1, Definition 3, this monic polynomial will be called the minimal polynomial of x over K.