Irreducibility of $1+X+\ldots+X^{p-1}$ over $\mathbb{Q}$

Question

Show that the polynomial:

$$g(X)=1+X+X^2+\ldots+X^{p-1}$$

is irreducible over $\mathbb{Q}$, where $p$ is prime.

I am not sure how to approach this.

Answer 1

Substitute $X=u+1$. $g(u)=g(X+1)=\dfrac{(X+1)^p-1}{X}=X^{p-1}+pX^{p-2}+\cdots +p$. Then use Eisenstein's criterion. The constant term $p$ is not divisible by $p^2$ though the other coefficients are divisible by $p$.