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fundamental theorem of algebra

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Theorem 1 Let $f \in \mathbb{C}[z]$ be a non-constant polynomial. Then there is a $z\in\mathbb{C}$ with .

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

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

"fundamental theorem of algebra" is owned by Mathprof. [ full author list (2) | owner history (1) ]

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Attachments:: proof of the fundamental theorem of algebra (Liouville's theorem) (Proof) by Evandar
proof of fundamental theorem of algebra (Proof) by scanez
fundamental theorem of algebra result (Theorem) by rspuzio
proof of fundamental theorem of algebra (due to D'Alembert) (Proof) by rspuzio
proof of fundamental theorem of algebra (argument principle) (Proof) by rspuzio
proof of fundamental theorem of algebra (Rouché's theorem) (Proof) by Wkbj79
polynomial equation of odd degree (Theorem) by pahio

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Cross-references: factors, algebraically closed, polynomial
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This is version 9 of fundamental theorem of algebra, born on 2002-02-13, modified 2006-12-03.
Object id is 1927, canonical name is FundamentalTheoremOfAlgebra.
Accessed 12994 times total.

Classification:

AMS MSC:	12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
	30A99 (Functions of a complex variable :: General properties :: Miscellaneous)

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