Let $F$ be a field, and let $f$ be a monic irreducible polynomial over $F$.
- Let $\alpha$ be a root of some other monic irreducible $g\ne f$. Then is $f$ still irreducible in $F(\alpha)$?
- Is it true that $f$ factors completely in $F(\alpha)$, where $\alpha$ is a root of $f$? Obviously $x-\alpha$ is a factor of $f$, but it is not clear to me that all the other roots are also in the field.
- Is $F(\alpha)$ isomorphic to $F(\beta)$ if $\alpha,\beta$ are roots of the same irreducible $f$? Are they equal (as subsets of some larger field that splits $f$)?
These are questions I have made up in order to better understand finite field extensions. I don't have a reference, and I believe all of them to be true, although I have no proof, which is why I am asking the question.
Edit: (Moved to a new question)