Fundamental theorem of algebra

The fundamental theorem of algebra, also called d'Alembert's theorem^[1] or the d'Alembert–Gauss theorem,^[2] states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.

Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.

The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division.

Despite its name, there is no purely algebraic proof of the theorem, since any proof must use some form of the analytic completeness of the real numbers, which is not an algebraic concept.^[3] Additionally, it is not fundamental for modern algebra; it was named when algebra was synonymous with the theory of equations.