Let $L/K$ be a galois extension of number field. Let $R_K$ and $R_L$ be ring of integers of $K$ and $L$. Let $p$ be a unratified prime of $K$. Let $P$ be a unique prime above $p$. There is map from $Gal(L_P/K_p)$ to $Gal((R_L/P)/(R_K/p))$.
I'm having trouble to prove this map is isomorphism. I understand the proof of subjectivity, but stuck with injectivity. I should have to use the condition $p$ is uramified. Thank you for your help.