= Solution
A <polynomial> over a <unique factorization domain> is called a <primitive polynomial> when the greatest common divisor of its coefficients is a unit.
Assume that the primitive <polynomial> $f$ factors in $F[X]$ as $f=gh$, with both factors of positive degree. Clearing denominators and removing contents writes
$$
g=a g_0,\qquad h=b h_0,
$$
where $a,b\in F^\times$ and $g_0,h_0\in R[X]$ are primitive. Then
$$
f=ab\,g_0h_0.
$$
By <Gauss lemma for polynomials>, $g_0h_0$ is primitive. Since $f$ is also primitive, comparison of contents forces $ab$ to be a unit of $R$. Absorbing that unit into one factor gives a nontrivial factorization of $f$ in $R[X]$, contrary to its assumed irreducibility. Hence $f$ is irreducible in $F[X]$.
Back to article page
