Content of Polynomials and Gauss's Lemma I am getting stuck on a little part of a proof: Let $R$ be a PID and let $K=$Frac$(R)$. If $f\in R[x]$ and $f=gh$ with $g\in R[x]$ of content 1, show that $h\in R[x]$. We can clear the denominators of $h$ (say by multiplying by $a$), to obtain $$ af=gh' $$ with $h'\in R[x]$. By Gauss's Lemma, we get that $$ a\cdot cont(f)=cont(g)cont(h')=cont(h'), $$ but I cannot see how to conclude that $h\in R[x]$.

The content of $h'$ is a multiple of $a$ which means that all the coefficients are multiples of $a$ so you can divide the coefficients of $h'$ by $a$ within $R$, and the result $h$ is a polynomial with coefficients in $R$.