Roots of different irreducible polynomials are algebraically independent Let $F$ be a field, and let $f$ be a monic irreducible polynomial over $F$.


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*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)

For your first problem, it need not be the case. For let $f(x)=x^4+16$. It is not hard to show that $f(x)$  is irreducible over the rationals. Note that
$$x^4+16=(x^2+2\sqrt{2} x+4)(x^2-2\sqrt{2} x+4),$$
and let $g(x)=x^2-2$.
