Computing the degree of the splitting field of $x^3+18x+3$ over $\Bbb Q$

Question

Let $T$ be a splitting field of polynomial $$f(x)=x^3+18x+3\in\mathbb{Q}[x].$$ What is the degree $[T:\mathbb{Q}]$?

My thoughts: the polynomial $f$ is irreducible over $\mathbb{Q}$, therefore the $[\mathbb{Q}(x_i):\mathbb{Q}]=3$, where $x_i$ is a root of $f$. Now I would need to get $[\mathbb{Q}(x_1,x_2):\mathbb{Q}(x_1)]$ and then $[\mathbb{Q}(x_1,x_2,x_3):\mathbb{Q}(x_1, x_2)]$. If $x_2, x_3$ were the complex roots, than $[\mathbb{Q}(x_1,x_2,x_3):\mathbb{Q}(x_1, x_2)]=1$.

So actually I need just $[\mathbb{Q}(x_1,x_2):\mathbb{Q}(x_1)]$. Any ideas?

Answer 1

If you denote $\alpha$ the real root of $f(x)$, you obtain the minimal polynomial of the complex roots of $f(x)$ dividing it by $x-\alpha$. It will have degree $2$ (actually it is $x^2+\alpha x+\alpha^2+18$), hence the degree of the splitting field over $\mathbf Q$ is $6$.

Answer 2

By the Rational Root Theorem, the only possible rational roots are $\pm 1, \pm 3$, but substituting shows that none of these are roots. Hence (since $\deg f < 4$), $f$ is irreducible, so $\operatorname{Gal}(f)$ is transitive and thus $3 \mid \#\operatorname{Gal}(f) = [T : \Bbb Q]$.

On the other hand, the discriminant of $f$ is $$\Delta_f = 4(18)^3 - 27(3^2) < 0 ,$$ so $f$ has two complex roots, and so complex conjugation is a nontrivial automorphism of $T / \Bbb Q$ of order $2$ , and hence $2 \mid \#\operatorname{Gal}(f)$. Since the order of the extension satisfies $[T : \Bbb Q] \mid (\deg f)! = 6$, we must have $[T : \Bbb Q] = 6$.

In fact, this generalizes to an efficient algorithm for computing the degree of the extension for any cubic (over $\Bbb Q$): If a cubic polynomial $g$ over $\Bbb Q$ is irreducible, then its splitting field is $\Bbb Q(\alpha, \sqrt{\Delta_g})$ for any root $\alpha$ of $g$ and either root $\sqrt{\Delta_g}$ of $\Delta_g$. Thus, $[T : \Bbb Q] = 3$ (and $\operatorname{Gal}(g) \cong A_3 \cong \Bbb Z_3$) if $\Delta_g$ is a square in $\Bbb Q$ and $[T : \Bbb Q] = 6$ (and $\operatorname{Gal}(g) \cong S_3$).