Exercise 1173Algebra › Rings and fieldsL1Added 3 Nov 2026

The polynomial ring $R[X]$

#polynomial ring#properties#integral domain

Statement

Show that if $R$ is an integral domain, then $R[X]$ is an integral domain, and the units of $R[X]$ are the units of $R$ .

Hint

Degree is additive: $\deg(PQ)=\deg P+\deg Q$ .

Show solutionHide solution

Integral domain. For nonzero $P,Q\in R[X]$ , $\deg P,\deg Q\ge0$ and $\deg(PQ)=\deg P+\deg Q\ge0$ , so $PQ\neq0$ .

Units. If $PQ=1$ , then $\deg P+\deg Q=0$ , so $\deg P=\deg Q=0$ . So $P,Q\in R$ and $PQ=1$ in $R$ .

Conclusion. $R[X]$ is an integral domain, and $R[X]^\times=R^\times$ .