I was wondering if someone knew a reference, that I could look up, of what I think is a fact that:
If $f(x), g(x)$ are two irreducible polynomials in $\mathbb{F}_p[x]$, for $p$ a prime, of respective degrees $n,m$. Then the splitting field of $h(x)=f(x)g(x)$ over $\mathbb{F}_p$ is simply $F_{p^d}$, where $d=lcm(m,n).$
I recall reading about this at some point, but I cannot seem to find a reference.
Thanks!