abstract algebra - Same minimal polynomial gives isomorphism - Mathematics Stack Exchange

Question

Let L and P be two field extensions of K. For $a in L$ and $bin P$ algebraic over K with the same minimal polynomial $f_a=f_b$ then there exists an isomorphism $w: K(a) ightarrow K(b)$ with $w(a)=b$.

Proof attempt: Since $f$ is irreducible and has $a$ as root then $K(a)$ is isomorphic to $K[x]/(f)$ with the isomorphism $X+f ightarrow a$. And we can do the same for $b$ and we get $K(b)$ is isomorphic to $K[x]/(f)$. Therefore $K(a)$ is isomorphic with $K(b)$.

My Questions:

1.What do I do now?

2."$f$ is irreducible and has $a$ as root then $K(a)$ is isomorphic to $K[x]/(f)$" we took it for granted in the lecture but why does this hold?

They might be trivial questions but I would really like to get an understanding of it.

Thanks in advance for the help.

Answer 1

Waitting for answers