Transitivity of integrality

Remarks

Let $c \in C$ and take a monic relation $c^n + b_{n-1} c^{n-1} + \cdots + b_0 = 0$ with $b_i \in B$. Each $b_i$ is integral over $A$, so $A' = A[b_0, \ldots, b_{n-1}]$ is a finitely generated $A$-module. The relation shows that $A'[c]$ is a finitely generated $A'$-module, so $A'[c]$ is a finitely generated $A$-module. Hence $c$ is integral over $A$ by the module criterion for integrality.

Example: $\sqrt[4]{2}$ is integral over $\Z[\sqrt{2}]$, as a root of $x^2 - \sqrt{2}$, and $\Z[\sqrt{2}]$ is integral over $\Z$. So $\sqrt[4]{2}$ is integral over $\Z$; indeed it is a root of $x^4 - 2$.

Going deeper

For a student

The proof does not build the monic polynomial over $A$ directly; it passes through finite generation of modules. The same argument shows that the elements of $B$ integral over $A$ form a subring, the integral closure of $A$ in $B$. Transitivity then shows that the integral closure is integrally closed in $B$.