Let $p$ be a prime number. $K$=$\mathbb C(x,y)$ and $F=\mathbb C(x^p,y^p)$.Then, Prove that $K/F$ is a Galois Extension.
Trial:
Since this $\mathbb C$ is a field of charactersitic $0$,it would be enough to show that $K$ is a splitting field of some separable polynomial. My guess is that the polynomial $f(t)=(t^p-x^p)(t^p-y^p)$.But i am not able to prove K is the splitting field of $f(t)$. Is this the right way and logic?
Yes, the suggested polynomial $f = (t^p - x^p)(t^p - y^p)\in \mathbb C(x^p,y^p)[t]$ does the job. Let $L$ be its splitting field.
Since $x$ and $y$ are roots of $f$, clearly $K \subseteq L$.
The full set of roots of $f$ is given by all $\zeta^i x$ and $\zeta^i y$ where $\zeta\in\mathbb C$ is a primitive $p$th root of unity and $i\in\{0,\ldots,p-1\}$. Since $x,y\in K$ and $\zeta\in\mathbb C\subseteq K$, all the roots of $f$ are contained in $K$ and thus $L\subseteq K$.
This shows $K = L$, so $K$ is the splitting field of $f$.