Definition:Transcendental (Abstract Algebra)/Field Extension

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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:

$\nexists \map f x \in F \sqbrk x \setminus \set 0: \map f \alpha = 0$

where $\map f x$ 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.