Isomorphism over $\mathbb{Q}$ but not isomorphism in $\mathbb{C}$

Question

Sorry for my bad English.

Is there example as follows?

$Let$ $K,L$ be subfield of $\mathbb{C}$ such that $L\neq K$ as subfield of $\mathbb{C}$, but $L\cong K$ as field (ignoring emmbding to $\mathbb{C}$) .

Answer 1

Take any $x,y\in \mathbb{C}$ which are transcendental over $\mathbb{Q}$. Then $\mathbb{Q}(x)\simeq \mathbb{Q}(y)$ are isomorphic to the field of rational fractions $\mathbb{Q}(X)$, but they are in general not equal.