Editing Algebraically closed field (section)

===Relatively prime polynomials and roots===
For any field ''F'', if two polynomials {{math|''p''(''x''), ''q''(''x'') ∈ ''F''[''x'']}} are [[coprime|relatively prime]] then they do not have a common root, for if {{math|''a'' ∈ ''F''}} was a common root, then&nbsp;''p''(''x'') and&nbsp;''q''(''x'') would both be multiples of {{math|''x'' &minus; ''a''}} and therefore they would not be relatively prime. The fields for which the reverse implication holds (that is, the fields such that whenever two polynomials have no common root then they are relatively prime) are precisely the algebraically closed fields.

If the field ''F'' is algebraically closed, let ''p''(''x'') and ''q''(''x'') be two polynomials which are not relatively prime and let ''r''(''x'') be their [[greatest common divisor]]. Then, since ''r''(''x'') is not constant, it will have some root ''a'', which will be then a common root of ''p''(''x'') and ''q''(''x'').

If ''F'' is not algebraically closed, let ''p''(''x'') be a polynomial whose degree is at least 1 without roots. Then ''p''(''x'') and ''p''(''x'') are not relatively prime, but they have no common roots (since none of them has roots).