If $a,b$ are transcendental over $\mathbb{Q}$, then it is known that $\mathbb{Q}(a)$ and $\mathbb{Q}(b)$ are isomorphic.
Consider a simple case: suppose $a,b,c$ are transcedental over $\mathbb{Q}$. Assume that $b\notin \mathbb{Q}(a)$ and $a\notin\mathbb{Q}(b)$. Consider than the extension $\mathbb{Q}(a,b)$. Can it be isomorphic to $\mathbb{Q}(c)$?
Note that I have put weaker condition on $a,b$; if $a,b$ are algebraically independent, then this is obvious that $b\notin \mathbb{Q}(a)$ and $a\notin\mathbb{Q}(b)$.
