Editing Fundamental theorem of algebra (section)

==Corollaries==
Since the fundamental theorem of algebra can be seen as the statement that the field of complex numbers is [[algebraically closed field|algebraically closed]], it follows that any theorem concerning algebraically closed fields applies to the field of complex numbers. Here are a few more consequences of the theorem, which are either about the field of real numbers or the relationship between the field of real numbers and the field of complex numbers:

* The field of complex numbers is the [[algebraic closure]] of the field of real numbers.
* Every polynomial in one variable ''z'' with complex coefficients is the product of a complex constant and polynomials of the form ''z''&nbsp;+&nbsp;''a'' with ''a'' complex.
* Every polynomial in one variable ''x'' with real coefficients can be uniquely written as the product of a constant, polynomials of the form ''x''&nbsp;+&nbsp;''a'' with ''a'' real, and polynomials of the form ''x''<sup>2</sup>&nbsp;+&nbsp;''ax''&nbsp;+&nbsp;''b'' with ''a'' and ''b'' real and ''a''<sup>2</sup>&nbsp;−&nbsp;4''b''&nbsp;<&nbsp;0 (which is the same thing as saying that the polynomial ''x''<sup>2</sup>&nbsp;+&nbsp;''ax''&nbsp;+&nbsp;''b'' has no real roots). (By the [[Abel–Ruffini theorem]], the real numbers ''a'' and ''b'' are not necessarily expressible in terms of the coefficients of the polynomial, the basic arithmetic operations and the extraction of ''n''-th roots.) This implies that the number of non-real complex roots is always even and remains even when counted with their multiplicity.
* Every [[rational function]] in one variable ''x'', with real coefficients, 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 real numbers), and rational functions of the form (''ax''&nbsp;+&nbsp;''b'')/(''x''<sup>2</sup>&nbsp;+&nbsp;''cx''&nbsp;+&nbsp;''d'')<sup>''n''</sup> (where ''n'' is a natural number, and ''a'', ''b'', ''c'', and ''d'' are real numbers such that ''c''<sup>2</sup>&nbsp;−&nbsp;4''d''&nbsp;<&nbsp;0). A [[corollary]] of this is that every rational function in one variable and real coefficients has an [[elementary function (differential algebra)|elementary]] [[Antiderivative|primitive]].
* Every [[algebraic extension]] of the real field is isomorphic either to the real field or to the complex field.