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Let $R$ be an integral domain containing a field $F$ as a subring. Show that if $R$ is a finite dimensional vector space over $F$, then $R$ is a field.

This is a Ph.D. entrance question, I recently appeared.

Somehow I need to prove that every element in $R$ has an inverse, but can't figure out how.

1 Answers1

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Hints:

$$(1)\;\;\;\forall\, r\in R\,\;\exists\,p(t)=a_0+a_1t+...+a_{n-1}t^{n-1}+t^n\in F[t]\;\;s.t.\;\;p(r)=0\;;$$

$$(2)\;\;\text{Lemma: let $\,A\subset B\,$ be two integral domains. If $\,B\,$ is integral over $\,A\,$ then} $$

$$\,A\, \text{ is a field iff $\,B\,$ is a field .}$$

DonAntonio
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