If the field has prime field isomorphic to $\mathbb{Q}$ would there be any subfield isomorphic to $\mathbb{Q}$?

Question

As title says, if the field has prime field isomorphic to $\mathbb{Q}$ would there be any other subfields isomorphic to $\mathbb{Q}$?

Answer 1

Let $K$ be a field and $Q$ be its prime field.

Let $L$ be a subfield of $K$. Then $L$ contains $Q$.

Every field homomorphism $Q \to L$ must be the inclusion because $Q$ is the prime field of $L$. Thus, the image of a field homomorphism $Q \to L$ is $Q$.

Therefore, $L$ is isomorphic to $Q$ iff $L=Q$.