Definition:Algebraic

Rings
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 $\alpha \in R$.

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

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

Fields
The same definition can be extended directly to fields:

Let $E / F$ be a field extension.

Let $\alpha \in E$.

Let $f \left({x}\right)$ be a polynomial in $x$ over $F$.

Then $\alpha$ is algebraic over $F$ if $\exists ~f \left({x}\right) \in F \left[{X}\right] - \{0\}$ such that $f \left({\alpha}\right) = 0$.

Field Extensions
A field extension $E / F$ is said to be algebraic if, $\forall ~\alpha \in E$, $\alpha$ is algebraic over $F$.

Transcendental
An element (or field extension) is said to be transcendental if it is not algebraic.