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[parent] zero polynomial (Definition)

The zero polynomial in a ring $ R[X]$ of polynomials over a ring $ R$ 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 $ R$ has no zero divisors, one may define that
$\displaystyle \deg(\textbf{0}) = -\infty$
or that the zero polynomial has no degree at all.

"zero polynomial" is owned by pahio.
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See Also: polynomial ring over integral domain, order and degree of polynomial, minimal polynomial (endomorphism)

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Cross-references: zero divisors, multiplication, absorbing element, polynomial ring, identity element, polynomials, ring
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This is version 8 of zero polynomial, born on 2004-10-29, modified 2006-06-04.
Object id is 6431, canonical name is ZeroPolynomial2.
Accessed 1689 times total.

AMS MSC12E05 (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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