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
 A: Partial answer: Let $f(x)=x^3+ax+b$, let $K$ be its splitting field, and $\alpha$, $\beta$ and $\gamma$ be the roots of $f$ in $K$.
First of all, $f$ has to be irreducible, which is the same as saying it doesn't have a rational root: if it's not, and say $\alpha$ is rational, then $f(x)$ factors as $(x-\alpha)g(x)$ with $g$ a quadratic polynomial; then $K$ is also the splitting field of $g$, so it has degree $\leq2$.
So let's assume that $f$ is irreducible. Its splitting field $\mathbf Q(\alpha)$ has degree $\deg(f)=3$, so if we want $[K:\mathbf Q]=3$, we need $K=\mathbf Q(\alpha)$. In other words, we want the two other roots $\beta,\gamma$ to be in $\mathbf Q(\alpha)$. Let's look at the relation between roots and coefficients for $f$:
$$
\alpha+\beta+\gamma=0\\
\alpha\beta+\beta\gamma+\gamma\alpha=a\\
\alpha\beta\gamma=-b
$$
From the first and the third equation, you see that $\beta+\gamma=-\alpha$ and $\beta\gamma=-b/\alpha=\alpha^2+a$, so $\beta$ and $\gamma$ are the roots of the second degree polynomial $g(y)=y^2+\alpha y +\alpha^2+a\in\mathbf Q(\alpha)$. Those roots are in $\mathbf Q(\alpha)$ if, and only if, the discriminant $\Delta=-3\alpha^2-4a$ of $g$ is a square in $\mathbf Q(\alpha)$.
Now, the problem is that I'm not sure how to determine whether $\Delta$ is a square or not...
A: 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]$.
