I came a cross a question that I don't know how to solve
Problem: $A,B$ are commutative domains and $A\subseteq B$. Show if that $B$ is a field and every element of $B$ is the root of a non-trivial monic polynomial in $A[x]$, then $A$ is a field.
I don't know how to start on the question.
Let $a \in A \setminus \{0\}$. By assumption, $a^{-1} \in B$ is well-defined and it is the root of a monic polynomial. Write down such an equation and multiply it with $a^n$, where $n$ is the degree of the polynomial. Now you will see the inverse of $a$ in $A$.