# Algebraic field extension and intermediate integral domain

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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Thanks!

• An algebraic field extension is not necessarily of finite degree; you cannot conclude that $\dim_KF=d$. It is also not necessary that $a^p\in D$ for any $p>0$. – Servaes Apr 8 '17 at 15:30
• $D = K[a_1,\ldots,a_n]$ where the $a_i$ are algebraic, and $K[a_1,\ldots,a_n]= K(a_1,\ldots,a_n)$ by the fundamental theorem $K[x]/(f(x)) \cong K[a]$ where $f$ is the minimal polynomial of $a$. – reuns Apr 8 '17 at 15:34
• Sorry I mistake algebraic extension with simple extension... – Ivon Apr 8 '17 at 15:37

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 $D$.
• It reduces to show that any $a \in F$ is algebraic over $K$ – reuns Apr 8 '17 at 15:41
• @Ivan : How would you show $a$ is algebraic ? – reuns Apr 8 '17 at 15:43