Integral dependence (also known as algebraic dependence) is a condition on ring extensions $R\subseteq S$: we say $s\in S$ is integral over $R$ if there is some $f(x) \in R[x]$ such that $f(s)=0$.

Integral dependence (also known as algebraic dependence) is a condition on ring extensions $R\subseteq S$: we say $s\in S$ is integral over $R$ if there is some $f(x) \in R[x]$ such that $f(s)=0$. The case where $R,S$ are fields forms the starting point of much of Galois theory.

An equivalent condition: $s\in S$ is integral iff there is a subring $T\subseteq S$ containing $s$ that is finitely-generated as an $R$-module. The idea is to use the (finite) powers of $s$ in $f(s)$ as generators.

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