Proving irreducibility of $x^6-72$

The problem becomes a lot simpler if you look at the field generated by a root of the polynomial. Let $\alpha^6 = 72$. Then $[{\mathbf Q}(\alpha):{\mathbf Q}] \leq 6$. Since $(\alpha^3/6)^2 = 2$ and $(\alpha^2/6)^3 = 1/3$, the field ${\mathbf Q}(\alpha)$ contains a square root of 2 and a cube root of 3. Thus $[{\mathbf Q}(\alpha):{\mathbf Q}]$ is divisible by 2 and by 3, hence by 6, so $[{\mathbf Q}(\alpha):{\mathbf Q}] = 6$, which is another way of saying the minimal polynomial of $\alpha$ over the rationals has degree 6. Therefore $x^6 - 72$ has to be the minimal polynomial of $\alpha$ over the rationals, so this polynomial is irreducible over $\mathbf Q$.

It is helpful but you can look at smaller primes as well.

Reduction modulo $5$ gives $$X^6 - 72 = (X^2 + 2)(X^2 + X + 2)(X^2 - X + 2).$$ Reduction modulo $7$ gives $$X^6 - 72 = (X^3 + 3)(X^3 - 3).$$

The reduction modulo $5$ implies that there is no factor of degree $3$, and the reduction modulo $7$ implies that there is no factor of degree $2$.

I had posted a more general question on Brilliant before, namely asking when is $p_n(x) = x^6 + n $ reducible over the integers (which is equivalent to reducible over the rationals as the content of the polynomial is 1.)

Suppose $p_n(x)=g(x)\cdot h(x),$ where $g$ and $h$ are not constants. The sum of the degrees of $g$ and $h$ is $6$, and the product of the leading coefficients is $1.$ Because all coefficients are integers, this means that the leading coefficients of $g$ and $h$ are either both $1$ or both $-1.$ In the latter case, we multiply both $g$ and $h$ by $-1$ so that the leading coefficients are $1$. Also, we can assume, without loss of generality, that $\deg(g)\geq \deg(h).$

The polynomial $p_n$ has $6$ complex roots, all with absolute value $\sqrt[6]{|n|}$. Suppose the degree of $h$ is $k$, which can be $1,2,$ or $3$. Then the absolute value of the free term of $h$ is the product of absolute values of $k$ roots, thus it is $|n|^{k/6}.$ If this is an integer, then $|n|$ must be either a perfect square (if $k=3$) or a perfect cube (if $k=2$) or a perfect 6th power (if $k=1$, though this is also a perfect cube). Moreover, if $k=3$, then $n$ cannot be positive, because every cubic polynomial has a real root, and $x^6+n>0$ for positive $n$.

Hence $p_n(x)$ is reducible if and only if $n = -a^2 $ or $b^3$.