Fundamental theorem of algebra

The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-<a href="/facts/Constant_polynomial/Lzak8VVx">constant</a> single-variable <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> with <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> has at least one complex <a href="/facts/Zero_of_a_function/mlK3dhoT">root</a>. This includes polynomials with real coefficients, since every real number is a complex number with its <a href="/facts/Imaginary_part/2w7mqI7e">imaginary part</a> equal to zero.
Equivalently (by definition), the theorem states that the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> is <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed</a>.
The theorem is also stated as follows: every non-zero, single-variable, <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> n polynomial with complex coefficients has, counted with <a href="/facts/Multiplicity_(mathematics)/dndUyetA">multiplicity</a>, exactly n complex roots. The equivalence of the two statements can be proven through the use of successive <a href="/facts/Polynomial_division/MgJlJ2G7">polynomial division</a>.
Despite its name, it is not fundamental for <a href="/facts/Modern_algebra/YNNE1CNz">modern algebra</a>; it was named when algebra was <a href="/facts/Synonym/a3BgMcdu">synonymous</a> with the <a href="/facts/Theory_of_equations/aUwHT1SD">theory of equations</a>.

Fundamental theorem of algebra open-in-new

Fundamental theorem of algebra