$\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ is a Galois extension, with $\mathbb{F}_4(x,y)$ a function field of an algebraic curve

Question

Consider the field extension $\mathbb{F}_4(x,y)/\mathbb{F}_4(x)$ where $y$ is a root of the polynomial $$ f(T)= x^4 + x^2T^2 + x^2T + x^2 + xT^2 + xT + T^4 + T^2 + 1 \in \mathbb{F}_4(x)[T]. $$ I want to show that $\mathbb{F}_4(x,y)/\mathbb{F}_4(x)$ is a Galois extension.

I already proved that $f(T)$ is separable. Now, I was thinking to find the splitting field of $f(T)$ and note that it is $\mathbb{F}_4(x,y)$. Is that the best way to solve this? How can I find the splitting field of $f(T)$? Any help is appreciated.

Answer 1

Notice that $f(T+1) = f(T+x) = f(T+x+1) = f(T)$.

This gives you explicitly the Galois group and proves that the extension is Galois.