Suppose $\gamma$ is the fifth root root of unity. That is, $\gamma = e^{\frac{2\pi i}{5}}$, so $\gamma$ is a root of $p(x) = x^5-1$, or more precisely of $x^4+x^3+x^2+x+1$ since we can factor out a $(x-1)$ which has the obvious root of $1$.
I need to show that $k = \mathbb{Q}(\gamma)$ is a splitting field for $p(x)$ over $\mathbb{Q}$.
I know I will probably have to consider the basis of this extension, which will be:
$\{1, \gamma, \gamma^2, \gamma^3, \gamma^4\}$
But I'm not exactly sure how to proceed. What I want to show is that my polynomial $p(x)$ can be factored into irreducible linear factors $(x-\alpha_1)(x-\alpha_2)...$ over $k$.
Because $(\gamma^n)^5 = (\gamma^5)^n = 1$, we have $$ x^5 - 1 = (x-1)(x-\gamma)(x-\gamma^2)(x-\gamma^3)(x-\gamma^4) $$ and so $x^5-1$ splits in $\mathbb{Q}(\gamma)$.