Let $A$ be a subring of a field $K$, and suppose that $A$ is a local ring with maximal ideal $\mathfrak{m}$. Let $x \in K, \, x \neq 0$. Let $\phi: A \rightarrow L$ be a homomorphism of $A$ into the algebraically closed field $L$ with kernel $\mathfrak{m}$. Suppose that $\mathfrak{m} A[x] \neq A[x]$. Then $\mathfrak{m} A[x]$ is contained into a maximal ideal $\mathfrak{P}$ of $A[x]$ and $\mathfrak{P} \cap A = \mathfrak{m}$. Now, there exists an embedding $\psi : A/ \mathfrak{m} \rightarrow L$ such that $A \rightarrow A/ \mathfrak{m} \stackrel{\psi}{\rightarrow} L$ is equal to $\phi$. Suppose that the . According to Lang's Algebra p. 348, we can extend $\psi$ to $A[x]/ \mathfrak{P}$ whether the image of $x$ in $A[x]/ \mathfrak{P}$ is transcendental over $A/ \mathfrak{m}$ or not, but i can not see why this is possible. Any insights?
Thanks
Edited:
Related to the question is the observation that, according to the proof of Theorem 5.21 in Atiyah's and McDonald's "Introduction to Commutative Algebra", where they follow a similar construction, it is mentioned that the image of $x$ in $A[x]/ \mathfrak{P}$ is algebraic over $A/ \mathfrak{m}$ (notation is different). But i can't see why that's the case.