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?