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

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

Let $\alpha \in E$.

Then $\alpha$ is transcendental over $F$ :
 * $\nexists \map f x \in F \sqbrk x \setminus \set 0: \map f \alpha = 0$

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.

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