TITLE: Quotient field of a certain quotient ring
QUESTION [6 upvotes]: Let $A$ be a commutative integral domain and $\mathfrak p$ a prime ideal of $A$. Let $A_{\mathfrak p}$ be the localization of $A$ at $\mathfrak p$ and $\mathfrak{m}_{\mathfrak{p}}=\mathfrak pA_{\mathfrak p}$ its unique maximal ideal. 
On page 10 of Serre's Local Fields, it states that $A_{\mathfrak p}/{\mathfrak{m}_{\mathfrak p}}$ is the field of fractions of the quotient ring $A/\mathfrak p$. 
Could someone help me prove this? 
In particular I'm unable to show that the residue field $A_{\mathfrak p}/{\mathfrak{m}_{\mathfrak{p}}}$ is contained in the quotient field of $A/\mathfrak p$.

REPLY [4 votes]: Here is another way to describe the situation; of course it is not so different to Keenan's.
There is an exact sequence $0 \to \mathfrak p \to A \to A/\mathfrak p \to 0.$  Localization is exact, and so localizing at $\mathfrak p$ gives an exact sequence
$$0 \to \mathfrak p A_{\mathfrak p} \to A_{\mathfrak p} \to (A/\mathfrak p)_{\mathfrak p} \to 0.$$  Thus it suffices to check that the localization of $A/\mathfrak p$ at $\mathfrak p$ is naturally identified with the fraction field of $A/\mathfrak p;$ but this is pretty clear, if you think about each in terms of adding certain denominators to the elements of $A/\mathfrak p$.