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 

You might as well make good use of the fact that the Galois group of an extension of finite fields is cyclic, generated by $x\mapsto x^q$, where $q$ is the cardinality of the smaller field. I think it’s clear that $\sqrt3+\alpha_1$ generates the whole degree-six extension $K$, and we can call this quantity $\rho$, so that everything in $K$ can be written uniquely in form $\sum_0^5n_i\rho^i$ with all $n_i\in\Bbb F_7$. The minimal polynomial for $\rho$ over $\Bbb F_7$ seems to be $2 -X + 2X^3 + X^6$. At any rate, your six automorphisms are $\rho\mapsto\rho^{7^m}$, $0\le m\le5$.
