# Definition: Polynomial Ring

The set of all polynomials. $p=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n}$

over the commutative ring $$R$$ (i.e. with $$a_{i}\in R,\,i=0,\ldots ,n,\,n\in \mathbb {N}$$), together with the component-wise addition and multiplication follow the usual rules of exponentiation $$X^{n}\cdot X^{m}:=X^{n+m}$$ is called polynomial ring and denoted by $$R[X]$$. Specifically, if $$p,q\in R[X]$$ with

$p=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{m}x^{m},$

and

$q=b_{0}+b_{1}x+b_{2}x^{2}+\cdots +b_{n}x^{n},$

then the addition of the two polynomials in $$R[X]$$ is defined as follows:

$p+q:=r_{0}+r_{1}x+r_{2}x^{2}+\cdots +r_{k}x^{k},\quad\quad k = \max(m, n),\,r_{i}:=a_{i}+b_{i}$

and

$p\cdot q:=s_{0}+s_{1}x+s_{2}x^{2}+\cdots +s_{l}x^{l},\quad\quad l = m + n,\,s_{i}=a_{0}b_{i}+a_{1}b_{i-1}+\cdots +a_{i-1}b_{1}+a_{i}b_{0}.$

Definitions: 1 2
Parts: 3

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### References

#### Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück