algebraically closed field

《Abstract Algebra》 · 第 543 页 · Section 13.4「Splitting Fields and Algebraic Closures」，Definition

A field $K$ is said to be algebraically closed if every polynomial with coefficients in $K$ has a root in $K$.

《Algebra》 · 第 285 页 · Section 5「Irreducibility of polynomials」，Definition 5.9、Lemma 5.10

A field $k$ is algebraically closed if all irreducible polynomials in $k[x]$ have degree $1$. […] A field $k$ is algebraically closed if and only if every nonconstant polynomial $f\in k[x]$ factors completely as a product of linear factors, if and only if every nonconstant polynomial $f\in k[x]$ has a root in $k$.