If $F$ is algebraic over $K$ and $D$ is an intermediate and an integral domain, is $D$ a field?

Question

I would imagine it would be as simple as showing that $D$ is finite since a finite integral domain is a field. I know that if $F$ is a finite extension of $K$, then $F$ is an algebraic extension of $K$. Is the converse true? That is, is the statement "if $F$ is an algebraic extension of $K$, then $F$ is a finite extension of $K$" true? That would in turn immediately imply that $D$ must be finite which concludes that $D$ is finite and a field.

Answer 1

I think you can reduce it to the finite case as follows. Let $a \in D$ be nonzero, and note that $a$ is algebraic over $K$. Hence, if $f(x)=x^n+c_{n-1}x^{n-1}+\dots+c_0$ is its minimal polynomial over $K$, then $$ -c_0a^{-1} = a^{n-1}+c_{n-1}a^{n-2}+\dots+c_1 \in D. $$ As $c_0$ is a unit in $D$, it follows that $a^{-1} \in D$. This shows that all nonzero elements of $D$ are units, hence it is a field.