Building splitting field over $\mathbb Q$ of $f(x) = x^{5} - 3$.
I know that the criterion of Eisenstein that $f(x)$ is irreducible with $p=3$.
Then $f(x)$ has five distinct roots. I know that $\alpha = \sqrt[5] 3$ is a root of $f(x)$.
But I'm having problem in finding other roots.
Hint: Take a $5$-th root of unit $\xi \neq 1$. Then let $\alpha = \sqrt[5]{3}$ and you have that $\alpha, \alpha\xi, \alpha\xi^2, \alpha\xi^3, \alpha\xi^4$ are the roots of $f(x) = x^5 - 3$.