Is $A/\varphi^{-1}(\mathfrak{m})\subseteq B/\mathfrak{m}$ an integral extension?

Question

Let $\varphi:A\rightarrow B$ be a homomorphism of finitely generated $k$-algebras, and let $\mathfrak{m}$ be a maximal ideal of $B$. We have the injective homomorphism $$ \overline{\varphi}:A/\varphi^{-1}(\mathfrak{m})\rightarrow B/\mathfrak{m}, a+\varphi^{-1}(\mathfrak{m})\mapsto \varphi(a)+\mathfrak{m}. $$ Is $A/\varphi^{-1}(\mathfrak{m})\subseteq B/\mathfrak{m}$ an integral extension?

Answer 1

Actually this is a finite extension of fields (hence algebraic, hence integral), by Zariski's Lemma.