Let $k$ be an algebraically closed field and $A$ a finitely generated $k$-algebra which is an integral domain. Let $m$ be a maximal ideal of $A$. Does the proof that $A_m/mA_m$ (i.e. the residue field of the localization of $A$ at $m$) is isomorphic to $A/m$ require anything deep like the Nullstelensatz? I.e. is there a basic ring theoretic argument that doesn't bring in any algebraic geometry or deeper results in commutative algebra? My difficulty is in proving the surjectivity of the map $A \rightarrow A_m/mA_m$. Thank you for your time.
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