Problem If $a$ and $b$ be any two conjugates over the field $F$, then there always exist an isomorphism $\psi: F(a)\to F(b)$ such that $\psi(a)=b$ and $\psi(k)=k$ for all $k\in F$.
We are given that $a$ and $b$ are conjugates over $F$ therefore they satisfy the same minimal polynomial say $p(x)$ over $F$. Then there exist an isomorphism
$\phi:F(a)\to F[x]/ \langle p(x)\rangle$ given by $\phi(k)=k+\langle p(x)\rangle$ and $ \phi(a)=x+\langle p(x)\rangle$
similarly, we can define a map $\phi^*$ $\phi^*:F(b)\to F[x]/\langle p(x)\rangle$ given by $\phi^*(k)=k+\langle p(x)\rangle$ and $ \phi^*(b)=x+\langle p(x)\rangle$.
$\phi^*$ is isomorphism.
Then there exist a composition map $\sigma:F(a)\xrightarrow{\psi} F[x]/\langle p(x)\rangle\xrightarrow{{\phi^*}^{-1}} F(b)$.
It's done.
Am I right? Any hint or help will be appreciable. Thanks!