I am trying to determine whether the polynomial $f(x) = x^6 + 34x^4 + 4x^2 + 89 \in \mathbb{Z}[x]$ is irreducible over $\mathbb{Z}$. Eisenstein's criterion doesn't help and I suppose I could determine whether $f$ is irreducible over $\mathbb{Z}/(p)$ for some prime $p$, but I could potentially have to compute the greatest common divisor of $f$ (reduced mod $p$) with $x^{p^6} - x$ and I am trying to avoid that for now.
My question is: If I was to let $u = x^2$, then $f$ becomes $u^3 + 34u^2 + 4u + 89$. If I was to show that $u^3 + 34u^2 + 4u + 89$ was irreducible over $\mathbb{Z}$, would that imply that $x^6+ 34x^4 + 4x^2 + 89$ is irreducible over $\mathbb{Z}$ as well?
Your polynomial is the square of $x^3-x-1$ over $\mathbb{F}_3$, hence it is, at best, the product of two irreducible polynomials over $\mathbb{Q}$, $a(x)$ and $b(x)$, both of them having degree $3$. However, $p(x)$ splits over $\mathbb{F}_7$ as $(x^2-3)(x^4+2x^2+3)$, and such factorizations are not compatible unless $p(x)$ is irreducible over $\mathbb{Q}$.
It is also interesting to point out that $89$ is a prime, hence $p(x)$ may split over $\mathbb{Q}$ only as $(x c(x)\pm 1)(x d(x)\pm 89)$. Moreover, $p(x)=p(-x)$ has to hold, hence it is not difficult to rule out every case.