A week ago, I came up with an injective, but not surjective homomorphism which mapped a field to the same field: $\phi: \Q(e) \to \Q(e)$ defined by $\phi(e) = e^2$ and $\left.\phi\right|_\Q = \id_{\Q}$. It isn’t surjective because $\phi[\Q(e)] = \Q(e^2) \subsetneq \Q(e)$

Obviously, this kind of mapping wasn’t defined on a finite field.

After that, I found another non-surjective embedding which sends field $\Z_p [y]$, where $y$ is an indeterminate, to itself on Mathematics Stack Exchange.¹

From this, I’ve understood that why the Isomorphism Extension Theomrem doesn’t apply to transcendental field extensions.

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