13-02 ALgebraic Extensions

Definition.

• $F$,
• $K$,
  • $F \subseteq K$,

$\alpha \in K$ is said to be algebraic over $F$ if

\[\exists f \in F[x] \text{ s.t. } f(\alpha) = 0 .\]

$\alpha \in K$ is said to be transcendental over $F$ if $\alpha$ is not algebraic over $F$. The extension $K/F$ is said to be algebraic if for every $\alpha \in K$ is algebraic over $F$.

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