Find conditions on $a$ and $b$ such that the splitting field of $x^3 +ax+b $ has degree of extension 3 Find conditions on $a$ and $b$ such that the splitting field of $x^3 +ax+b \in \mathbb Q[x]$ has degree of extension 3 over $\mathbb Q$.
I'm trying solve do this question, it seems very difficult to me, maybe because I don't know Galois Theory yet (The content of next chapters). I need help to prove this without Galois Theory. 
Thanks

The following conditions should give you a splitting field of degree 3 over $\mathbb{Q}$:


*

*$a,b \in \mathbb{Z}$.

*There exists prime $p$ such that $p \mid a$ and $p \mid b$.

*$p^2 \nmid b$.


Then by Eistenstein's criterion, then the polynomial is irreducible over $\mathbb{Q}[x]$ and therefore $$\big [ \mathbb{Q}(\alpha) \colon \mathbb{Q} \big ] = \textrm{Deg}(x^3 + ax + b) = 3.$$
You may be able to get a more general result by considering the case where the coefficients (again, $a,b \in \mathbb{Z}$) are coprime ($\textrm{gcd}(a,b) = 1$) and applying Gauss' lemma.
Lemma: A non-constant polynomial $p(x)$ is irreducible over $\mathbb{Z}[x]$ if and only if it is irreducible over $\mathbb{Q}[x]$ and primitive over $ \mathbb{Z}[x]$.
