How can I find the Galois group of x^5 + 2x +2?
If you reduce mod the polynomial factors as . Sorry if this method is really computational, this is the only method I was able to think of.
Therefore by Dedekind's theorem the Galois group contains a product of a 2-cycle and 3-cycle. The only transitive subgroup of which contains a product of a 2-cycle and a 3-cycle is itself. Therefore, the Galois group is .