Ring of Polynomials

definitionpolynomials

Definition

For a commutative ring $R$ with unity, the set of polynomials $R[x]=\{P(x)\mid P(x)\text{ polynomials}\}$ is a ring with coefficient-wise addition, and multiplication (the same as in the group algebra for $G=(\mathbb{Z},+)$) given by the following properties,

1. $x^i{x}^j=x^{i+j}$.
2. Follows commutativity and distributivity.

Properties

• $\mathrm{deg}(P+Q)\le \max (\mathrm{deg}P,\mathrm{deg}Q)$ (is strict if $LT(P)=-LT(Q)$).
• If $R$ is a domain (and $P,Q\ne 0$), $\mathrm{deg}(PQ)=\mathrm{deg}P+\mathrm{deg}Q$.
• If $R$ is a domain, then $R[x]$ is a domain.

Examples