# zero polynomial

The in a ring $R[X]$ of polynomials over a ring $R$ is the identity element 0 of this polynomial ring:

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

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.

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

Because always

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

and because in general  $\deg(fg)=\deg(f)+\deg(g)$  when $R$ has no zero divisors, one may define that that the zero polynomial has no degree (http://planetmath.org/Polynomial) at all, or alternatively that

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

(see the extended real numbers).

Title zero polynomial ZeroPolynomial 2013-03-22 14:46:58 2013-03-22 14:46:58 pahio (2872) pahio (2872) 13 pahio (2872) Definition msc 13P05 msc 11C08 msc 12E05 PolynomialRingOverIntegralDomain OrderAndDegreeOfPolynomial MinimalPolynomialEndomorphism