# 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 subdomain of $R$.

Let $x \in R$.

Then $x$ is **algebraic over $D$** if and only if:

- $\exists f \left({x}\right)$ over $D$ such that $f \left({x}\right) = 0$

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**.

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 64 \ (2)$