An exercise states to (try to) prove that the polynomial $x^6-72$ is irreducible over the rationals.
As $72=2^33^2$, we cannot apply Eisenstein criterion without any additional trick. I was wondering if there is a neat way to show that this polynomial is irreducible? The only thing that comes to my mind is to deal with undefined coefficients and then solve polynomial systems, but maybe there is a more conceptual way?
[That there are no linear factors is obvious. Reductions modulo $2$ and $3$ give $X^6=0$. Reduction modulo $p=73$ gives (partial) factorization $X^6+1=(X^2+1)(X^4-X^2+1)$, but I am not sure if this information is of any help here.]
