If $F\subseteq K$ are fields, $\alpha \in K$, and $K$ is an extension field of $F$.
Prove the following are equivalent:
- $\alpha$ is algebraic over $F$
- $|F(\alpha):F|$ is finite
I'm trying to prove the properties from here: https://en.wikipedia.org/wiki/Algebraic_element
Edited the question because people voted to close for not understanding what I mean in the original question:
Intuition: I did prove (2.) and (3.) independently, but I'm not sure how to show (1.)