Definition:Transcendental (Abstract Algebra)/Field Extension
< Definition:Transcendental (Abstract Algebra)(Redirected from Definition:Transcendental 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.