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# fundamental theorem of algebra

###### Theorem.

Let $f\in\mathbb{C}[Z]$ be a non-constant polynomial. Then there is a $z\in\mathbb{C}$ with $f(z)=0$.

In other words, $\mathbb{C}$ is algebraically closed.

As a corollary, a non-constant polynomial in $\mathbb{C}[Z]$ factors completely into linear factors.

Related:

ComplexNumber, Complex, TopicEntryOnComplexAnalysis, ZeroesOfDerivativeOfComplexPolynomial

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

12D99*no label found*30A99

*no label found*

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## Attached Articles

proof of the fundamental theorem of algebra (Liouville's theorem) by Evandar

proof of fundamental theorem of algebra by scanez

fundamental theorem of algebra result by rspuzio

proof of fundamental theorem of algebra (due to d'Alembert) by rspuzio

proof of fundamental theorem of algebra (argument principle) by rspuzio

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proof of fundamental theorem of algebra (due to Cauchy) by pahio

proof of fundamental theorem of algebra by scanez

fundamental theorem of algebra result by rspuzio

proof of fundamental theorem of algebra (due to d'Alembert) by rspuzio

proof of fundamental theorem of algebra (argument principle) by rspuzio

proof of fundamental theorem of algebra (Rouch\'e's theorem) by Wkbj79

polynomial equation of odd degree by pahio

proof of fundamental theorem of algebra (due to Cauchy) by pahio