## Finite Extensions

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.
Q.E.D.

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?