Let $F/K$ be an algebraic field extension. Suppose $D$ is an integral domain with $K\subset D\subset F$. Show that $D$ is a field.
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2026-03-26 17:43:45.1774547025
Algebraic field extension and intermediate integral domain
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If $a\in D\setminus\{0\}$ then $D$ contains the $K$-algebra $K[a]$, and because $a$ is algebraic over $K$ we have $K[a]=K(a)$, therefore $a$ has an inverse in $D$.