# Commutative Algebra – Generalization of the Atiyah-Macdonald Proposal 5.7

The proposal is

To let $$A$$ $$subseteq$$ $$B$$ be holistic domains, $$B$$ all over $$A$$, Then $$B$$ is a field iff $$A$$ is a field.

The proof is simple. I would like to generalize this proposal. I want to prove that

To let $$A to$$ $$B$$ be holistic domains, $$B$$ all over $$A$$, Then $$B$$ is a field iff $$A$$ is a
Field.

One page is simple: Suppose $$A$$ is a field, let $$y in B, y not = 0$$, Since $$B$$ is all over $$A$$, To let $$y ^ {n} + f (a_ {1}) y ^ {n-1} + … + f (a_ {n}) (a_ {i} in A)$$ be an equation of integral dependence for y of the least 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. But I do not know how to prove the other side: if $$B$$ So it's a field $$A$$ is a field.

Can you tell me how to prove it?