Definition:Algebraic Element of Field Extension
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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 $F \sqbrk X \to E$ 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 $F$.
Also see
- Definition:Algebraic Field Extension
- Definition:Algebraic Independence
- Definition:Integral Element of Algebra