# What is the Galois group of a polynomial over a finite field?

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• What is the Galois group of the splitting field of $X^8-3$ over $\mathbb{Q}$?

I've computed the splitting field of $x^8-3$ over $\mathbb{Q}$ to be $\mathbb{Q}(\sqrt[8]{3},\zeta_8)=\mathbb{Q}(\sqrt[8]{3},\sqrt{2},i)$, which is of degree 32 over $\mathbb{Q}$. The possible automorphisms are the maps fixing $\mathbb{Q}$ of form $$\sqrt[8]{3}\mapsto \zeta_8^i\sqrt[8]{3}\quad (0\leq i\leq 7),\qquad \sqrt{2}\mapsto\pm\sqrt{2},\qquad i\mapsto\pm i.$$ There are 32 automorphisms, and thus these are all automorphisms. So I have an explicit description of the automorphisms in the...

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Your description of $G$ is perfectly fine as it is. But maybe a representation of $G$ in \$GL_2(\mathbb...

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