Let $f \in \mathbb{Q}[x]: f(x) = x^3 + p x + q, p \geq 0, f$ irreducible therein; $\mathbb{k}$ be the splitting field of $f$ over $\mathbb{Q}$; $G = Gal(\mathbb{k} / \mathbb{Q})$. I have to show that $G \cong S_3$. I have to prove it algebraically and using information from the pictoral graph of $f$.
I have already proven some results. I have shown that $\exists!$ field $\mathbb{E}: \mathbb{Q} \subseteq \mathbb{E} \subseteq \mathbb{k}, [\mathbb{E}:\mathbb{Q}]=2, \mathbb{E}=\mathbb{Q}(\sqrt{D})=\mathbb{k}^H$, where $D = -4p^3-27q^2$ and $H \leq G: H \cong A_3$. I also proved that $A_3 \cong H = G \implies \sqrt{D} \in \mathbb{Q}$.