My book gave a problem, If are fields and can be regarded as a finite extension over . And where is an integral domain. Then prove that is a field.
This is how I proved it, first is a commutive ring with unity, thus if it can be shown that every element is a unit in then it is a field.
(The obvious way to prove this problem).
1)If then .
2) is finite extension over .
3)Thus is algebraic extension over .
4)Thus, is finite extension over
5)Thus has inverse because it is a field.
7)Thus, since and are closed.
8)Thus, is a division ring.
But my problem is that the fact that has no zero divisors was never used in the proof. Thus, this is true for any commutative ring with unity.
Am I missing something?