Show that $R$ is a field.

Question

let $E$ be a splitting field of a polynomial over $k$. Let $R$ be a ring such that $k\subset R \subset E$ Then $R$ is a field.

Does not it follow from the fact that any integral domain which is finite dimensional vector space over a field is a field.

Is there any other solution of this?

Answer 1

Another way to see it is to observe that $E/k$ is algebraic. If $a\in R\setminus \{0\}$, then also $a\in E$, and hence $a$ is algebraic over $k$. Therefore $k(a) = k[a] \subseteq R$, and in particular $a^{-1}\in R$. Consequently, $R^\times = R\setminus\{0\}$ and $R$ is a field.