Compute the Galois group of $p(x)=x^4+ax^3+bx^2+cx+d$

Compute the Galois group of the following polynomial: $$p(x)=x^4+ax^3+bx^2+cx+d$$

Step 1: Calculate cubic resolvent

Step 2: Calculate the discrimimant $\gamma$ of the cubic resolvent

Step 3: If $\gamma$ is a square then the Galois group is $A_4$, else $S_4$

Is this correct for all $a, b, c, d \in \mathbb{Z}$?

How do we know if the Galois group is, for example, $\mathbb{Z_4}$?

• As you recognize, there are other possibilities than just $S_4$ and $A_4$, so your Step 3 must be wrong. For examples, we have $f=X^4-4X^2-2$ and $g=X^4+X^3-6X^2-X+1$. Both of them give normal extensions with Galois group $Z_4$. You can check irreducibility of $g$ by calculating $g(X+4)$. – Lubin May 31 '16 at 3:59
• Thanks. What further step can we take to distinguish $\mathbb{Z}_4$ from $S_4$ and $A_4$? – amiz9 May 31 '16 at 4:04
• And $\mathbb{Z_2} \times \mathbb{Z_2}$ is another abelian possibility I believe.. – amiz9 May 31 '16 at 4:07
• Sorry, this is not my game at all; all I would do is thrash around inefficiently till I found the automorphisms. There must be algorithms, but I don’t know them. – Lubin May 31 '16 at 4:15
• Another possibility if the dihedral group $D_8$, which is the Galois group of $x^4-2$, for example. There is a discussion of algorithms here: mathoverflow.net/questions/22923 – Derek Holt May 31 '16 at 7:41