Definition:Transcendental (Abstract Algebra)/Field Extension

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Definition

A field extension $E / F$ is said to be transcendental if and only if:

$\exists \alpha \in E: \alpha$ is transcendental over $F$

That is, a field extension is transcendental if and only if it contains at least one transcendental element.


Transcendental Element

Let $E / F$ be a field extension.

Let $\alpha \in E$.


Then $\alpha$ is transcendental over $F$ if and only if:

$\nexists f \left({x}\right) \in F \left[{x}\right] \setminus \left\{{0}\right\}: f \left({\alpha}\right) = 0$

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


Also see

If no element of $E / F$ is transcendental over $F$, then $E / F$ is algebraic.


Historical Note

The term transcendental, in the sense of meaning non-algebraic, was introduced by Gottfried Wilhelm von Leibniz.