TITLE: Ring Inside an Algebraic Field Extension
QUESTION [8 upvotes]: Let $E|F$ be an algebraic field extension and a ring $K$ such that $F\subseteq K\subseteq E$. It is true that $K$ is a field?

REPLY [3 votes]: A more general argument can be given: let $A\subset B$ be two integral domains. An element $b\in B$  is said to be integral over $A$ if there exists a monic polynomial $p(t)\in A[t]$ with $p(b)=0$. The ring $B$ is said to be integral over $A$ is every element in $B$ is integral over $A$.
Theorem: Let $A\subset B$ be I.D.'s , with $B$ integral over $A$. Then, $A$ is a field iff $B$ is a field.