---- > [!definition] Definition. ([[algebraically closed]]) > A [[field]] $k$ is **algebraically closed** if all [[irreducible element of an integral domain|irreducible]] [[polynomial 4|polynomials]] in $k[x]$ have degree $1$. ^definition > [!equivalence] > The following are equivalent: > - $k$ is algebraically closed; > - Every nonconstant polynomial $f \in k[x]$ factors completely as a product of linear (i.e., degree-1) factors; > - Every nonconstant polynomial $f \in k[x]$ has a [[root of a polynomial|root]] in $k$. ^equivalence ---- #### ---- #### 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 > ```