Why '$Aut((R_L/P)/(R_K/p))$ is galois' is not obvious? Let $L/K$ be a number field abelian extension which is finite. Let fix a prime ideal $p$ of $K$ and $P$ be a prime ideal of $L$ above $p$.
Let $R_L$ and $R_K$ be ring of integers of $L$ and $K$.
　Then there is surjective homomorphism from decomposition group $D_p$ to corresponding residue field extension.
　Almost every book discusses the reason why $Aut((R_L/P)/(R_K/p))$ is galois.
But to me, $R_K/p$ is finite field and $[ R_L/P:R_K/p]$ is residue degree and is finite. So the extension is galois because any finite extension of finite field is galois.
　So to me,$Aut((R_L/P)/(R_K/p))$ is galois is obvious, why books discusses hardly this issue ? Where am I missing ?

Books may discuss it because they are interested in a more general case, like a Galois extension of fraction fields of integrally closed domains, not just a Galois extension of number fields.  Are you really finding a book that spends time on that mapping where the residue fields are only assumed to be finite?
Example. Lang's Algebraic Number Theory discusses this result for a finite Galois extension $L/K$ of the fraction field $K$ of an integrally closed domain $A$ and its integral closure $B$ in $L$. If the residue field extension is not separable (this can really happen), then the residue feld extension can be normal but not separable and thus is not a Galois extension. This would be a good reason to write ${\rm Aut}((B/\mathfrak P)/(A/\mathfrak p))$ for the automorphisms of the residue field extension.
When you are only interested in a special case, complicated arguments from the general case may greatly simplify, but that doesn't mean the more general case is not worthwhile merely because you don't have any interest in it yet.
