Galois group of a splitting field of a polynomial over $\mathbb{F}_7$ Question: Find the Galois group of the splitting field $(x^2-3)(x^3+x+1)$ over $\mathbb{F}_7$. 
I know the splitting field is $K:=\mathbb{F}_7(\sqrt{3},\alpha_1)$, where $\alpha_1$ is one of the roots of the polynomial $x^3+x+1$. I know that the possible automorphisms of K fixing F must have the mappings $\sqrt{3} \mapsto \pm\sqrt{3}$ and $\alpha_1 \mapsto \{\alpha_1,\alpha_2,\alpha_3\}$ where $\alpha_1,\alpha_2,\alpha_3$ are the distinct roots of $x^3+x+1$. But when I wanted to write out all the automorphsism explicitly, I have some trouble. Any help will be appreciated 

Note that the Galois group is some subgroup of the direct product of the Galois groups of each factor considered individually. Since the splitting field of $x^2 - 3$ over $\Bbb{F}_7$ has degree two, the splitting field of $x^3 +x+1$ has degree three, and the degrees are coprime the splitting field of their product has degree 6. The direct product of the Galois groups of the factors, $\Bbb{Z}_2 \times \Bbb{Z}_3$, has order 6, and the Galois group of $K$ is a 6 element subgroup of this so it must be the whole group. If you want it explicitly, a generator is the permutation $\sigma$ sending $\sqrt{3}$ to its negative and sending $\alpha _1 \to \alpha _2 \to \alpha_3 \to \alpha _1$. This is necessarily an automorphism, because the Galois group acts on the $\alpha _i$ as the alternating group $A_3$.
