Why can't Eisenstein's Criterion be used to show that $$4x^{10} - 9x^{3} + 21x - 18$$ is irreducible over $\mathbb{Q}$?
I mean even if we were to apply Eisenstein here, there doesn't exist a prime $p$ that would apply all the E.C. rules anyways.
A detailed explanation would be great! Thanks.
The only prime that divides 9, 21, and 18 is 3. But $3^2 | 18$, so the Eisenstein criterion does not apply here.