Exegesis. diagram illustrations and a special case [1408A]
General case
$F_{1} = F_{2} = F, K_{1} = K_{2} = K$ implies that if $\alpha, \beta \in K$ are both roots of an irreducible polynomial $p(x) \in F[x]$, then
$$
F(\alpha) \cong F(\beta), \quad a(\alpha) \mapsto a(\beta), \quad a(x) \in F[x]
$$
is well-defined and is an isomorphism.
Special case
Example 1. An example for $F=\mathbb{Q}$ and $K=\mathbb{C}$ [1408AC]
