My book gave a problem, If F,E are fields and E can be regarded as a finite extension over F. And F\subseteq D\subseteq E where D is an integral domain. Then prove that D is a field.

This is how I proved it, first D is a commutive ring with unity, thus if it can be shown that every element is a unit in D then it is a field.
(The obvious way to prove this problem).

1)If x\in D then x\in E.
2) E is finite extension over F.
3)Thus E is algebraic extension over F.
4)Thus, F(x) is finite extension over F
5)Thus x\in F(x) has inverse x^{-1} because it is a field.
6)Thus, x^{-1}=a_0+a_1x...+a_nx^n,a_i\in F.
7)Thus, x^{-1}\in D since a_i,x\in D and D are closed.
8)Thus, D is a division ring.

But my problem is that the fact that D has no zero divisors was never used in the proof. Thus, this is true for any commutative ring with unity.
Am I missing something?