Definition:Transcendental Element of Field Extension

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:

$\nexists f \left({x}\right) \in F \left[{x}\right] \setminus \left\{{0}\right\}: f \left({\alpha}\right) = 0$

where $f \left({x}\right)$ denotes a polynomial in $x$ over $F$.


Also 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.