Let $(A, \mathfrak{m})$ be a local finite $k$-algebra, where $k$ is a field. Let $\Omega $ be a algebraically closed extension of $k$. Then for each $k$-algebra homomorphism $g: A \to \Omega$, does there exists $f : A/\mathfrak{m} \to \Omega$ such that $g = f \circ \pi$, where $\pi : A \to A/\mathfrak{m}$ is the natural surjection ? Or is there any other similar (general ) theorem ?
In fact, I now trying to prove next statement ( C.f. Q.1) in Understanding the Proposition 12.18 in Gortz's Algebraic Geometry ) : "Let $k \subset \Omega$ be a field extension where $\Omega$ is algebraically closed. Let $(A , \mathfrak{m})$ be a local finite $k$-algebra with residue field $\kappa(x)$. Then $[\kappa(x) : k]_{sep}=\#\operatorname{Hom}_k(A, \Omega)$ ( $[\kappa(x) : k]_{sep}$ is the separable degree)."
At least we know that $[\kappa(x) : k]_{sep}=\#\operatorname{Hom}_k(\kappa(x), \Omega) = \#\operatorname{Hom}_k(A/\mathfrak{m}, \Omega) $ ( C.f. N. Bourbaki, Algebra, Chapter 4-7, V. 7.9 , or it is mentioned in the Gortz, Wedhorn, Algebraic Geometry, Remark B.99 ). So it is sufficient to show that $\#\operatorname{Hom}_k(A/\mathfrak{m}, \Omega)= \#\operatorname{Hom}_k(A, \Omega)$. As a first trial, let's define $\phi : \operatorname{Hom}_k(A/\mathfrak{m}, \Omega) \to \operatorname{Hom}_k(A, \Omega) $ by $f \mapsto f \circ \pi$, where $\pi : A \to A/\mathfrak{m}$ is the natural surjection. Since $\pi$ is an epimorphism, $\phi$ is injective. Then can we show the surjectivity? Our above question ( extendibility ) is true? Or perhaps, my original question ( $[\kappa(x) : k]_{sep}=\#\operatorname{Hom}_k(A, \Omega)$ ) which I am trying to prove is false?
