PlanetMath: zero of polynomial

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zero of polynomial

(Definition)

Let be a subring of a commutative ring . If is a polynomial in , it defines an evaluation homomorphism from to . Any element $\alpha$ of satisfying

$\displaystyle f(\alpha) = 0$

is a zero of the polynomial

If also is equipped with a non-zero unity, then the polynomial is in divisible by the binomial $x\!-\!\alpha$ (cf. the factor theorem).

For example, the real number $\sqrt{2}$ ( $\in \mathbb{R}$ ) is a zero of the polynomial $X^2\!-\!2$ of the polynomial ring $\mathbb{Q}[X]$ .

"zero of polynomial" is owned by pahio.

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Also defines:

zero of the polynomial

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Cross-references: polynomial ring, real number, factor theorem, binomial, divisible, non-zero unity, evaluation homomorphism, polynomial, commutative ring, subring
There are 4 references to this entry.

This is version 3 of zero of polynomial, born on 2008-08-27, modified 2008-08-28.
Object id is 10962, canonical name is ZeroOfPolynomial.
Accessed 377 times total.

Classification:

AMS MSC:	12E05 (Field theory and polynomials :: General field theory :: Polynomials )
	11C08 (Number theory :: Polynomials and matrices :: Polynomials)
	13P05 (Commutative rings and algebras :: Computational aspects of commutative algebra :: Polynomials, factorization)

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