Is $X^8+3X^4-53$ irreducible over $\mathbb{Z}[X]$

Question

I want to determine whether or not the polynomial $X^8+3X^4-53$ is irreducible over $\mathbb{Z}[X]$. I noticed that it doesn't have integer (or rational) roots but I have no further ideas.

Answer 1

Following criterion of Osada can be applied:

Let $f(x)=x^n+a_1x^{n-1}+\dots+a_{n-1}x\pm p$ be a polynomial with integer coefficients, where $p$ is a prime. If $p>1+|a_1|+\dots+|a_{n-1}|$, then $f$ is irreducible.

Conditions are satisfied by $p=53$ and $p>1+3$.

The criterion can be found for example as Theorem 2.2.7 in Prasolov's book Polynomials.