## commutative algebra – Generalization of the Atiyah-Macdonald Proposition 5.7

The proposal is

Let $$A$$ $$subseteq$$ $$B$$ to be integral domains, $$B$$ integral on $$A$$. then $$B$$ is a field so $$A$$ is a field.

The proof is easy. I want to generalize this proposition. I want to prove that

Let $$A to$$ $$B$$ to be integral domains, $$B$$ integral on $$A$$. then $$B$$ is a field so $$A$$ is a
field.

One side is easy: suppose $$A$$ is a field, let $$y in B, y not = 0$$. Since $$B$$ is an integral part of $$A$$, let $$y ^ {n} + f (a_ {1}) y ^ {n-1} + … + f (a_ {n}) (a_ {i} to A)$$ an integral dependence equation for y of the smallest possible degree. then $$a_n not = 0$$, so $$y ^ {- 1} = – f (a_ {n} ^ {- 1}) (y ^ {n-1} + … + f (a_ {n-1})) in B$$, Therefore $$B$$ is a field.However, I do not know how to prove the other side: if $$B$$ is a field, then $$A$$ is a field.

Could you tell me how to prove it?