Exercise 3.14

Prove that if $R$ is an integral domain, then $R[x]$ is also an integral domain.

Answer

Let $R$ be an integral domain. Now, consider $R[x]$. Clearly, $0\ne 1$. Now, let $a,b\in R[x]$ such that $ab=0$. Suppose both $a$ and $b$ are nonzero, then $a=a_0+a_{1}x+\cdots +a_n{x}^n$ and $b=b_0+b_{1}x+\cdots +b_m{x}^m$ where $\mathrm{deg}(a)=n$ and $\mathrm{deg}(b)=m$. Then, the coefficient of $x^{m+n}$ in $ab$ is $a_n{b}_m$. However, $ab=0$. Therefore $a_n{b}_m=0$. Since $R$ is an integral domain, it must be that $a_n=0$ or $b_m=0$, but the leading coefficient of a polynomial can’t be zero. Therefore, our assumption is false and thus, $a=0$ or $b=0$. Therefore, $R[x]$ is also an integral domain. $■$