What does it mean when one talks about splitting field of a multivariable polynomial? And then, Galois group of that splitting field?

Question

I came across the following, while I was reading a recent research article. I do not know how to interpret it; the article does not define this concept (perhaps because it is too elementary). Could anyone possibly help me or point me in an appropriate direction? The article reads:

Let $F(x,y)$ be an irreducible integral binary cubic form having splitting field with Galois group $S_{3}$?

In my basic Galois theory class, we never considered splitting field of a polynomial in two variables. Since there are two variables, splitting field is an extension of what field? Rationals?

Answer 1

Here's an example . . .

Let $F(x,y)=x^3-2y^3$, and let $f(x)=F(x,1)=x^3-2$.

Then letting $a,b,c$ denote the roots of $f$, we get $$ F(x,y) = y^3f\Bigl({\small{\frac{x}{y}}}\Bigr) = y^3 \Bigl({\small{\frac{x}{y}}}-a\Bigr) \Bigl({\small{\frac{x}{y}}}-b\Bigr) \Bigl({\small{\frac{x}{y}}}-c\Bigr) = (x-ay)(x-by)(x-cy) $$ so the splitting field of $F$ over $\mathbb{Q}$ is $\mathbb{Q}(a,b,c)$, which is the same as the splitting field of $f$ over $\mathbb{Q}$.

Since $[\mathbb{Q}(a,b,c):\mathbb{Q}]=6$, the Galois group of $f$ is $S_3$.