The field extension \({A}\Big/{\mathfrak m}\) over \(k\) is finite hence algebraic. Since \(k\) is algebraically closed, \({A}\Big/{\mathfrak m}\) is isomorphic to \(k\).

Existence: the kernel of \(A \stackrel{\pi }{\to } {A}\Big/{\mathfrak m} \stackrel{\sim }{\to } k\) is exactly \(\mathfrak m\).

Uniqueness: assume \(\phi \) and \(\psi \) are two \(\mathbb {C}\)-algebra homomorphism such that \(\mathfrak {m}=\ker \phi = \ker \psi \). Let \(\rho : \mathbb {C}\to R\) be the structure map of \(R\). For any \(r \in R\), we claim that there exists some \(x \in \mathfrak m\) and \(\lambda \in \mathbb {C}\), such that \(r = x + \rho (\lambda )\). Indeed, we have that \(r = (r - \rho (\phi (r))) + \rho (\phi (r))\), and \(r - \rho (\phi (r))\) is in the kernel of \(\phi \). Thus for every \(r = m + \rho (\lambda )\), we have \(\phi (r) = \phi (\rho (\lambda )) = \psi (\rho (\lambda )) = \psi (r)\).