Given a proper field extension $L/K$, can we have $L\cong K$? [duplicate]
Given a proper field extension $L/K$ (that is, $K$ can be considered as a proper sub-field of $L$). Can it still happen that $K\cong L$ via a field-isomorphism? I assume No, but I am utterly illiterate in basic field theory, so if this is really easy, then I would already be happy with a hint in the right direction.
Solution 1:
I suppose you can take $K$ to be a function field on countably many indeterminates, and get $L$ by adding one more indeterminate.