Irreducibility of the cyclotomic polynomial

Shift. $\Phi_p(X+1)=\frac{(X+1)^p-1}{X}=\sum_{k=1}^p\binom pk X^{k-1}=X^{p-1}+pX^{p-2}+\cdots+p$.

Eisenstein. The polynomial $\Phi_p(X+1)$ has leading coefficient $1$ and all other coefficients divisible by $p$, with constant term $p$ not divisible by $p^2$. By Eisenstein's criterion, $\Phi_p(X+1)$ is irreducible over $\mathbb{Q}$.

Conclusion. $\Phi_p(X)$ is irreducible over $\mathbb{Q}$ (irreducibility is preserved by the shift $X\mapsto X+1$).