My line of proof is as follows:
- $\pm1$ are the only candidates for being rational roots (Rational Root Theorem)
- Since $f(1)=5$ and $f(-1)=1$, none is a root
And hence the given polynomial is irreducible over the rationals $\mathbb{Q}$.
Is this proof correct? I have doubt because the solution given in the book uses Eisenstein's criterion by modifying the polynomial to $f(x+1)=\frac{(x+1)^5-1}{x}$ and so on... which seems complicated to me.