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fundamental theorem of algebra

(Theorem)

Theorem 1 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.

"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
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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 16694 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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