Editing Algebraically closed field (section)

===Every polynomial is a product of first degree polynomials===
The field ''F'' is algebraically closed if and only if every polynomial ''p''(''x'') of degree ''n''&nbsp;≥&nbsp;1, with [[coefficient]]s in ''F'', [[factorization|splits into linear factors]]. In other words, there are elements ''k'',&nbsp;''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;...,&nbsp;''x<sub>n</sub>'' of the field ''F'' such that ''p''(''x'')&nbsp;=&nbsp;''k''(''x''&nbsp;&minus;&nbsp;''x''<sub>1</sub>)(''x''&nbsp;&minus;&nbsp;''x''<sub>2</sub>)&nbsp;⋯&nbsp;(''x''&nbsp;&minus;&nbsp;''x<sub>n</sub>'').

If ''F'' has this property, then clearly every non-constant polynomial in ''F''[''x''] has some root in ''F''; in other words, ''F'' is algebraically closed. On the other hand, that the property stated here holds for ''F'' if ''F'' is algebraically closed follows from the previous property together with the fact that, for any field ''K'', any polynomial in ''K''[''x''] can be written as a product of irreducible polynomials.