---- > [!definition] Definition. ([[algebraic field extension]]) > Let $k \subset F$ be a [[field extension]], and let $\alpha \in F$. Then $\alpha$ is **algebraic over $k$, of degree $n$** if $n=[k(\alpha): k]$ is finite; $\alpha$ is **transcendental over $k$** otherwise. > The extension $k \subset F$ is **algebraic** if every $\alpha \in F$ is algebraic over $k$. ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```