Definition:Transcendental (Abstract Algebra)/Field Extension/Element

From ProofWiki
Jump to navigation Jump to search

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

Special cases


Historical Note

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