# Irreducibility of $X^p -p$

Is the polynomial $X^p -p$ irreducible in $\mathbb{Q}(\zeta)$ where $\zeta$ is a primitive root of unity? I think i can't use eisentein criterion in the usual way... so how cain proceed?

• is $\zeta$ a $p$th root of unity or are you asking the question for any $n$th root of unity? – hunter May 8 '18 at 23:20
• @hunter i'm sorry is a $p$-th root of unity – andres May 9 '18 at 7:18

## 1 Answer

Assuming $\zeta$ is a primitive $p$th root of unity, yes. We can assume $p \neq 2$ since the result is obvious for $X^2 - 2$ over $\mathbb{Q}$. Then the degree of $\mathbb{Q}(\zeta)$ over $\mathbb{Q}$ is $p-1$, and in particular is coprime to $p$.

Since $X^p - p$ is irreducible over $\mathbb{Q}$ (Eisenstein), the degree of $\mathbb{Q}(\sqrt[p]{p})$ over $\mathbb{Q}$ is $p$. It follows from the multiplicativity of degrees in extensions that the degree of $\mathbb{Q}(\zeta, \sqrt[p]{p}$) over $\mathbb{Q}(\zeta)$ must be $p$.

• and it is enought to say that the polynomial is irreducible over the cyclotomic field? – andres May 9 '18 at 7:44