PlanetMath: zero polynomial

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zero polynomial

(Definition)

The zero polynomial in a ring of polynomials over a ring is the additive identity element $\textbf{0}$ of this polynomial ring:

$\displaystyle f+\textbf{0} = \textbf{0}+f = f \quad\forall\, f\in R[X]$

So the zero polynomial is also the absorbing element for the multiplication of polynomials.

All coefficients of the zero polynomial are equal to 0, i.e.

$\displaystyle \textbf{0} := (0,\,0,\,0,\,...).$

Because always

$\displaystyle f\cdot\textbf{0} = \textbf{0}$

and because in general $\deg(fg) = \deg(f)+\deg(g)$ when

has no zero divisors, one may define that that the zero polynomial has no degree at all, or alternatively that

$\displaystyle \deg(\textbf{0}) = -\infty$

"zero polynomial" is owned by pahio.

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Cross-references: extended real numbers, zero divisors, coefficients, multiplication, absorbing element, polynomial ring, identity element, polynomials, ring
There are 10 references to this entry.

This is version 9 of zero polynomial, born on 2004-10-29, modified 2008-06-03.
Object id is 6431, canonical name is ZeroPolynomial2.
Accessed 2911 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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