How to determine Galois group of the $X^6-10X^3+1$ over rational field $\mathbb Q$?
My thoughts: Firstly, $X^6-10X^3+1$ is irreducible over Q, and then do we need to compute all the roots of this polynomial so that we can get its Galois Group? I am learning graduate-level algebra right now, but I am not sure how to analyze these kinds of question. Can someone tell me how to solve this problem?