Editing Complex number (section)

===Fundamental theorem of algebra===
The [[fundamental theorem of algebra]], of [[Carl Friedrich Gauss]] and [[Jean le Rond d'Alembert]], states that for any complex numbers (called [[coefficient]]s) {{math|''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub>}}, the equation
<math display=block>a_n z^n + \dotsb + a_1 z + a_0 = 0</math>
has at least one complex solution ''z'', provided that at least one of the higher coefficients {{math|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>}} is nonzero.<ref name="Bourbaki 1998 loc=§VIII.1">{{harvnb|Bourbaki|1998|loc=§VIII.1}}</ref> This property does not hold for the [[rational number|field of rational numbers]] <math>\Q</math> (the polynomial {{math|''x''<sup>2</sup> − 2}} does not have a rational root, because {{math|√2}} is not a rational number) nor the real numbers <math>\R</math> (the polynomial {{math|''x''<sup>2</sup> + 4}} does not have a real root, because the square of {{mvar|x}} is positive for any real number {{mvar|x}}). 

Because of this fact, <math>\Complex</math> is called an [[algebraically closed field]]. It is a cornerstone of various applications of complex numbers, as is detailed further below. 
There are various proofs of this theorem, by either analytic methods such as [[Liouville's theorem (complex analysis)|Liouville's theorem]], or [[topology|topological]] ones such as the [[winding number]], or a proof combining [[Galois theory]] and the fact that any real polynomial of ''odd'' degree has at least one real root.