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

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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.