Same minimal polynomial gives isomorphism 
Let L and P be two field extensions of K. For $a \in L$ and $b\in P$ algebraic over K with the same minimal polynomial $f_a=f_b$ then there exists an isomorphism $w: K(a) \rightarrow K(b)$ 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 $X+f\rightarrow a$. 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.

Consider the homomorphism
\begin{align}
K[X]&\longrightarrow L,\\
P(X)&\longmapsto P(a).
\end{align}
Its image is $K(a)$ and its kernel is the ideal $(f)$, whence, by the First Isomorphism theorem,
$$K[X]/(f)\simeq K(a),$$
and similarly for $K(b)$.
