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.