Example of a field embedding from K to K which is not an Automorphism.

Question

A field embedding from K to K is an injective homomorphism. What is an example of a field embedding which is not surjective (and therefore not an automorphism)?

Answer 1

Take a field $K$ and consider $K(x_1,x_2,\ldots)$, the field of rational functions in countably infinite variables with coefficients in $K$ (equivalently, a transcendental field extension of $K$ with countably infinite transcendence degree, and no non-trivial algebraic elements).

Now consider the embedding $$ K(x_1,x_2,\ldots)\to K(x_1,x_2,\ldots)\\ 1\mapsto 1\\ x_i\mapsto x_{i+1} $$