I am reading the book abstract Algebra and the book claims that if $p(x)$ is irreducible polynomial over a field $F$ and $g(x)$ is polynomial of smaller degree then $\gcd(p,g)$ is invertible.
for example: $F=\mathbb{Q}$, $p(x)=x^3-2$ and $g(x)=x+1$ $\implies$ $\gcd(p,g)=1$
Why is this true ?