Definition:Algebraic Element of Field Extension

Definition

Let $E / F$ be a field extension.

Let $\alpha \in E$.

Definition 1

$\alpha$ is algebraic over $F$ if and only if it is a root of some nonzero polynomial over $F$:

$\exists f \in F \sqbrk X \setminus \set 0: \map f \alpha = 0$

where $F \sqbrk X$ denotes the ring of polynomial forms in $X$.

Definition 2

$\alpha$ is algebraic over $F$ if and only if the evaluation homomorphism $F \sqbrk X \to K$ at $\alpha$ is not injective.

Degree

Let $E / F$ be a field extension.

Let $\alpha \in E$ be algebraic over $F$.

The degree of $\alpha$ is the degree of the minimal polynomial $\map {\mu_F} \alpha$ whose coefficients are all in $F$.