Definition:Transcendental (Abstract Algebra)/Field Extension/Element
< Definition:Transcendental (Abstract Algebra) | Field Extension(Redirected from Definition:Transcendental Element of Field Extension)
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Definition
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 known as
- The phrase transcendental over $F$ can also be seen as transcendental in $F$. Both forms are used on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Also see
- Definition:Transcendental Field Extension
- Definition:Algebraic Element of Field Extension: If $\alpha \in E$ is not transcendental over $F$ then it is algebraic over $F$.
Special cases
Historical Note
The term transcendental, in the sense of meaning non-algebraic, was introduced by Gottfried Wilhelm von Leibniz.