Corollary 1

Let A be an integral domain, K its field of fractions and K' a finite-dimensional K-algebra (not necessarily commutative). If x \in K' is integral over A, the coefficients of the characteristic polynomial Pc_{K'/K}(x; X) (Algebra, Chapter VIII, § 12, no. 2) are integral over A.

If z \mapsto M(z) is the regular representation of the algebra K' (considered as a matrix representation; cf. Algebra, Chapter VIII, § 13) Pc_{K'/K}(x; X) is by definition the characteristic polynomial of the matrix M(x); if x integral over A, the matrix M(x) is integral over A (no. 1, Proposition 2) and it is sufficient to apply Proposition 17.