A generalization of the Eisenstein criterion

Recently, in the Zulip server for Lean users, somebody went with something that looked like homework, but managed to sting me a little bit. It was about irreducibility of polynomials with integer coefficients. Specifically, the guy wanted a proof that the polynomial $T^4-10 T^2+1$ is irreducible, claiming that the Eisenstein criterion was not good at it. What was to proven (this is what…