Algebraic extension

Definition

Let $R\le S$ be commutative unital rings. We say that $S$ is an algebraic extension of $R$ if for any $a\in S$, there exists a polynomial $p(x)\in R[x]$ such that $p(a)=0$.

The term algebraic extension is typically used for an algebraic extension of integral domains, or an algebraic field extension

Relation with other properties

Stronger properties

• Integral extension
• Separable algebraic extension
• Normal algebraic extension