Editing Algebraically closed field (section)

===The only irreducible polynomials are those of degree one===
The field ''F'' is algebraically closed if and only if the only [[irreducible polynomial]]s in the [[polynomial ring]] ''F''[''x''] are those of degree one.

The assertion "the polynomials of degree one are irreducible" is trivially true for any field. If ''F'' is algebraically closed and ''p''(''x'') is an irreducible polynomial of ''F''[''x''], then it has some root ''a'' and therefore ''p''(''x'') is a multiple of {{math|''x'' &minus; ''a''}}. Since ''p''(''x'') is irreducible, this means that {{math|1=''p''(''x'') = ''k''(''x'' &minus; ''a'')}}, for some {{math|''k'' ∈ ''F'' \&thinsp;{0} }}. On the other hand, if ''F'' is not algebraically closed, then there is some non-constant polynomial ''p''(''x'') in ''F''[''x''] without roots in ''F''. Let ''q''(''x'') be some irreducible factor of ''p''(''x''). Since ''p''(''x'') has no roots in ''F'', ''q''(''x'') also has no roots in ''F''. Therefore, ''q''(''x'') has degree greater than one, since every first degree polynomial has one root in ''F''.