How to find the Galois group of a polynomial? I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I  seem to be having some trouble with the technical stuff. A specific example would be how to find the Galois group of a given polynomial. I know some tricks, and I manage to solve some of those questions, but some not.
For example, one of the tests asks to find the Galois group of $x^{4}-4x+2$.
I can see that it is irreducible over $\mathbb Q$ (Eisenstein), but I have no clue as to how to find its Galois group over $\mathbb Q$. Can someone tell me how to do this? General techniques concerning this sort of problems are also welcome :).
Thanks!
 A: The Galois group of a (irreducible) polynomial is a transitive permutation group of its roots. How many real roots does it have? What can you say about non-real roots? What are the transitive subgroups of $S_4$? For those, see http://hobbes.la.asu.edu/Groups/group-data/degree4.html from Transitive Group Data.
Consider the splitting field. Since adding a real root of the polynomial gives you an extension degree of 4 and that there are non-real roots, the degree of the splitting extension is at least 8, ie, is a multiple of 8. So, the Galois group is either $D_8$ (the dihedral group of order 8) or $S_4$.
A: There are standard tests for polynomials of degree $3$ and $4$, and beyond that there are strategies that sometimes work and sometimes don't. There is a fairly thorough discussion in Keith Conrad's expository papers here (degrees $3$ and $4$) and here (Galois groups $S_n$ and $A_n$). 
For irreducible quartic polynomials, there are five possible Galois groups: $S_4, A_4, D_4, V_4, C_4$. The polynomial $x^4 - 4x + 2$ has one extreme point, so has either $2$ or $0$ real roots, and by inspection it has $2$. That means that complex conjugation is an element of the Galois group, and it acts as a transposition. 
Edit, 6/27/15: The original answer here contained an error; at this point we know that the Galois group is either $S_4$ or $D_4$, since those are the two possibilities that contain a transposition. To resolve this ambiguity you can use the techniques described in Keith Conrad's papers above or here. 
