I have only learned Galois theory over rational numbers, $\mathbb{Q}$, in my algebra and/or algebraic number theory class, which helps study polynomials $f(x)$ over one-variable by studying their splitting fields. I had a question to see if we could study splitting fields of multivariable polynomial analogously.
If $f(x,y)$ is a homogeneous polynomial of two variables then, we could find a polynomial $F(x) \in\mathbb{Z}[x]$ such that $f(x,y) = y^{m} F(\frac{x}{y})$. Here $m$ is the degree of $f(x,y)$. Then we could find splitting of $F(z) = (a_{1}z + b_{1}) \cdots (a_{m}z + b_{m})$ into linear factors and its splitting field. Note that $f(x,y) = (a_{1}x + b_{1}y) \cdots (a_{m}x + b_{m}y)$. Does it make sense say that the splitting field of $f(x,y)$ over $\mathbb{Q}(y)$ is $\mathbb{Q}(y)(a_{1}, \ldots , a_{m}, b_{1}, \ldots, b_{m})$, ignoring the $a_{j}, b_{j}$ that are already rationals?
If the answer to above is YES, how can we do this for non-homogeneous polynomial? If the answer to above is NO, is there any way to study the splitting field of multivariable polynomials?
I would also appreciate a lot if someone could point me to any relevant references.