I know that a finite field extension $E/F$ is integral. I think the converse is not true. Somebody told me it was true under the hypothesis that $E$ is a finitely-generated $F$-module. I've thought about this for about an hour and I am stuck. Maybe somebody can lend a hint?
closed as off-topic by Rob Arthan, user26857, Namaste, Leucippus, hardmath Feb 23 '18 at 4:55
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – user26857, Namaste, Leucippus
Let $\alpha \in E$. If $E$ is a finitely-generated $F$-module, then $E$ is a finite-dimensional vector space over $F$ and so is $F(\alpha)$. This implies that $\alpha$ is integral over $F$.