I am currently maintaining a Julia software package CommutativeRings which implements also factorization of integer polynomials according to Bareiss and Zassenhaus. If the polynomial is of the form $f(x^{a})$ where $a > 1$ it checks first, if $f(x^{d})$ is reducible for some divisor $d | a$ to exploit this information. The question arose, if it is necessary to check all divisors, or if it would be sufficient to consider for example prime factors of $a$.
If p and q are two different prime numbers, and $f(x)$ is a polynomial in $\mathrm{Z}[x]$. Is it true or is there a counter-example for the following statement:
If $f(x^p)$ is irreducible and $f(x^q)$ is irreducible, then $f(x^{p q})$ is irreducible.
Or more general for $f(x^{p q r})$ for any integer $r$.
I tried a long list of generated irreducible polynomials $f$ of degrees up to $10$, and $a = 6$ without finding a counter-example.