Find the condition on $a, b$ such that the field of $x^3+ax+b\in\Bbb{Q}[x]$ has degree of extension 6.

Question

First of all the polynomial $p(x)=x^3+ax+b$ must be irreducible over $\Bbb{Q}$, because if not then the degree of extension of its splitting field will be $1$ or $2$.
Now, suppose $\alpha, \beta, \gamma$ be three roots of $p(x)$.
Then I observe if we have $[\Bbb{Q}(\alpha):\Bbb{Q}]=3, [\Bbb{Q}(\beta):\Bbb{Q}]=2$ and $\gamma\in\Bbb{Q}(\alpha, \beta)$, then we are done. Because in that case we have $[\Bbb{Q}(\alpha, \beta):\Bbb{Q}]=6$(since $\operatorname{gcd}(3,2)=1$)
But in this manner can I reach the condition on $a$ and $b$?
Can anybody help me in this regard? Thanks for assistance in advance.
N.B. I have solved the problem when the degree of extension of its splitting field is $3$.