$E$ is a splitting field of $f(x)=x^3-3x+1$ over $\mathbb Q$.Then determine the group $G(\frac {E}{\mathbb Q})$
MY try : Actually I don't Know how to relate splitting field to group,But I found something like $Gal(E/\mathbb{Q}) \simeq S_3$
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3The Galois group is $A_3$ since the discriminant is a square in $\mathbb{Q}$ – Lozenges Mar 30 '17 at 07:46
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@Lozenges Why not expand a bit and post it as an answer? – Arthur Mar 30 '17 at 07:49
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$f(x)$ is the minimal polynomial of $2\cos\frac{2\pi}{9}$, whose algebraic conjugates are the real numbers $2\cos\frac{4\pi}{9}$ and $2\cos\frac{8\pi}{9}$. By the cosine duplication formulas the map $\alpha\mapsto 2\left(\frac{\alpha}{2}\right)^2-1$ is a generator of $\text{Gal}(\mathbb{E}/\mathbb{Q})$ and the last Galois group is isomorphic to $\mathbb{Z}/(3\mathbb{Z})$.
Jack D'Aurizio
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