Definition:Transcendental 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.

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.