Definition:Algebraic Element of Ring Extension

Definition
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $\left({D, +, \circ}\right)$ be an integral domain such that $D$ is a subring of $R$.

Let $\alpha \in R$.

Then $\alpha$ is algebraic over $D$ iff:
 * $\exists f \left({x}\right)$ over $D$ such that $f \left({\alpha}\right) = 0$

where $f \left({x}\right)$ is a non-null polynomial in $x$ over $D$.

Also see

 * An element of $R$ is said to be transcendental if it is not algebraic.