Editing Fundamental theorem of algebra (section)

{{Short description|Every polynomial has a real or complex root}}
{{Distinguish|Fundamental theorem of arithmetic|Fundamental theorem of linear algebra}}
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The '''fundamental theorem of algebra''', also called '''d'Alembert's theorem'''<ref>{{citation|url=https://old.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020748.02p0019l.pdf|title=Euler and the fundamental theorem of algebra|first=William|last=Dunham|journal=The College Journal of Mathematics|volume=22|issue=4|pages=282–293|date=September 1991|doi=10.2307/2686228 |jstor=2686228}}</ref> or the '''d'Alembert–Gauss theorem''',<ref>{{citation|url=http://www.math.toronto.edu/campesat/ens/20F/14.pdf|title=14 - Zeroes of analytic functions|work=MAT334H1-F – LEC0101, Complex Variables|publisher=University of Toronto|first=Jean-Baptiste|last=Campesato|date=November 4, 2020|access-date=2024-09-05}}</ref> states that every non-[[constant polynomial|constant]] single-variable [[polynomial]] with [[Complex number|complex]] [[coefficient]]s has at least one complex [[Zero of a function|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 (mathematics)|field]] of [[complex number]]s is [[Algebraically closed field|algebraically closed]].

The theorem is also stated as follows: every non-zero, single-variable, [[Degree of a polynomial|degree]] ''n'' polynomial with complex coefficients has, counted with [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial|multiplicity]], exactly ''n'' complex roots. The equivalence of the two statements can be proven through the use of successive [[polynomial division]].

Despite its name, it is not fundamental for [[modern algebra]]; it was named when algebra was [[synonym]]ous with the [[theory of equations]].