Let $x,y$ be variables. Let $a,b,c,d\in\mathbb{Z}$ and $n=|ad-bc|$. Show that $L=\mathbb{C}(x,y)$ is a Galois extension of $K=\mathbb{C}(x^ay^b,x^cy^d)$ of degree $n$. Find $\text{Gal}(L/K)$.
What I need to show here is $|\text{Aut}(L/K)|=[L:K]=n$, i.e. number of automorphisms from $L$ to itself, keeping $K$ as fixed field, equals the extension degree $n$. But $n=|ad-bc|$ and those powers in $x,y$ in the subfield $K$ is confusing me to prove that equality above. Also I'm unable to construct the Galois group of $L$ over $K$. Any help is appreciated.
I would look for automorphisms of the form $(x,y)\mapsto(\zeta^rx,\zeta^sy)$ where $\zeta=\exp(2\pi i/n)$. These are always automorphisms of $L=\Bbb C(x,y)$ and fix all the elements of $K$ iff $$ar+bs\equiv cr+ds\equiv0\pmod n.\tag1$$ I reckon that $(1)$ has $n$ distinct solutions $(r,s)$ modulo $n$, giving you all the automorphisms you need.