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

Definition
Let $E / F$ be a field extension.

Let $\alpha \in E$.

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

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.

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

 * Definition:Transcendental Number