Quickest way to prove irreduciblity?

Question

Is there any quick way to show that $x^6-3$ is irreducible over $\Bbb F_7$ without using rabin's test ?

Answer 1

It's irreducible if a root generates the degree 6 extension $\mathbb{F}_{7^6}$

A relatively easy way to figure out what field that roots generate is that if $\alpha$ is a primitive $n$-th root of unity, then the $m$-th roots of $\alpha$ are primitive $mn$-th roots of unity. So you just need to find the smallest extension capable of having a primitive $mn$-th root of unity.