Definition:Algebraic

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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 $\alpha \in R$.

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 $f \left({\alpha}\right) = 0$

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 $F$ iff:

$\exists f \left({x}\right) \in F \left[{X}\right] - \left\{{0}\right\}: f \left({\alpha}\right) = 0$

A field extension $E / F$ is said to be algebraic iff:

$\forall \alpha \in E: \alpha$ is algebraic over $F$

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

Algebraic number, which is the application of the definition of algebraic over $\Q$