How can I find the Galois group of x^5 + 2x +2?
If you reduce mod $\displaystyle 29$ the polynomial factors as $\displaystyle (x^2+4x+7)(x^3+25x^2+9x+21)$. 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 $\displaystyle S_5$ which contains a product of a 2-cycle and a 3-cycle is $\displaystyle S_5$ itself. Therefore, the Galois group is $\displaystyle S_5$.