---- > [!definition] Definition. ([[finite field extension]]) > A field extension $k \subset F$ is said to be **finite of degree $n$** if $F$ has (finite) dimension $\text{dim }F=n$ as a [[vector space]] over $k$. It is **infinite** otherwise. The degree of a finite extension is denoted $[F : k]$ (and we write $[F:k]=\infty$ if the extension is infinite). ^definition > [!generalization] > This is [[finite algebra|finite extension]] in the special case where both [[ring|rings]] are in fact [[field|fields]]. ^generalization ---- #### ---- #### 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 > ```