section_path: Definition and basic operations/Fundamental theorem of algebra
section_name: Fundamental theorem of algebra
paragraph_text: 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 coefficients) a,..., a, the equation a n z n + ⋯ + a 1 z + a 0 = 0 {\displaystyle a_{n}z^{n}+\dotsb +a_{1}z+a_{0}=0} has at least one complex solution z, provided that at least one of the higher coefficients a,..., a is nonzero. This property does not hold for the field of rational numbers Q {\displaystyle \mathbb {Q} } (the polynomial x − 2 does not have a rational root, because √2 is not a rational number) nor the real numbers R {\displaystyle \mathbb {R} } (the polynomial x + 4 does not have a real root, because the square of x is positive for any real number x).
  
  Because of this fact, C {\displaystyle \mathbb {C} } 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, or 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.