$E$ be an algebraic extension of $F$. Suppose $K$ is an integral domain s.t. $F \leq K \leq E$. Show that $K$ is a field.

Question

$E$ be an algebraic extension of $F$. Suppose $K$ is an integral domain s.t. $F \leq K \leq E$. Show that $K$ is a field.

I've been struggling with this one for awhile now! Anyone got any insights? Thanks!

Answer 1

Hint:

Note the hypothesis ‘$K$ is an integral domain` is redundant. Being a subalgebra is enough.

Let $x\in K$, $x\ne o$. You have to show $x$ has an inverse in $K$. Consider the subalgebra $L=E[x]$ and the map: $\begin{aligned}[t] m_x:L&\longrightarrow L,\\ y&\longmapsto xy. \end{aligned}$

This is a linear map, which is injective, and $L$ is a finite dimensional $E$-vectorspace. What can you conclude?