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Suppose that $K$ is an algebraic extension of a field $F$. Suppose that $R$ is a ring with $F\subseteq R\subseteq K$. I'm trying to prove that $R$ is a field. But I have no idea. Anyone can help me?

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Suppose $x \in R$. If $x \in F$, then $x$ has an inverse and we are done. Otherwise $x \in K$, and therefore $a_nx^n + ... + a_1x=1$ for some $a_i \in F$ (modify the satisfying polynomial suitably). Now, $x(a_nx^{n-1} + ... + a_1) = 1$, whence $x$ has an inverse in $R$. It follows that $R$ is a field.

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