Editing Algebraically closed field (section)

===Decomposition of rational expressions===
The field ''F'' is algebraically closed if and only if every [[rational function]] in one variable ''x'', with coefficients in ''F'', can be written as the sum of a polynomial function with rational functions of the form ''a''/(''x''&nbsp;−&nbsp;''b'')<sup>''n''</sup>, where ''n'' is a natural number, and ''a'' and ''b'' are elements of ''F''.

If ''F'' is algebraically closed then, since the irreducible polynomials in ''F''[''x''] are all of degree 1, the property stated above holds by the [[Partial fraction decomposition#Statement of theorem|theorem on partial fraction decomposition]].

On the other hand, suppose that the property stated above holds for the field ''F''. Let ''p''(''x'') be an irreducible element in ''F''[''x'']. Then the rational function 1/''p'' can be written as the sum of a polynomial function ''q'' with rational functions of the form ''a''/(''x''&nbsp;–&nbsp;''b'')<sup>''n''</sup>. Therefore, the rational expression
:<math>\frac1{p(x)}-q(x)=\frac{1-p(x)q(x)}{p(x)}</math>
can be written as a quotient of two polynomials in which the denominator is a product of first degree polynomials. Since ''p''(''x'') is irreducible, it must divide this product and, therefore, it must also be a first degree polynomial.