Q:

Prove that a primitive $q$-th root of unity is in the algebraic closure of $\Bbb F_p$

Let $p$ and $q$ be odd primes. Let $\Omega$ be the algebraic closure of $\Bbb F_p$. Let $\omega$ be a primitive $q$-th root of unity. Show that $\omega \in \Omega$.

How do I show that? Please help me in this regard.
Thank you very much.

A:

By definition $\omega$ is a root of $X^q-1\in\Bbb{F}_p[X]$, and by definition every polynomial in $\Bbb{F}_p[X]$ splits into linear factors in $\Omega[X]$. Hence $X-\omega\in\Omega[X]$ and so $\omega\in\Omega$.

