On the morphism induced by inclusion of fields in Hartshorne II 6.10

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This was a question bothered me since II 4. I would like to ask explicitly how can a inclusion of fields induce a morphism?

Meanwhile, regarding the application of Proposition 6.8, we will need the fact that the image space is the whole space. How can we show that the image of the designated morphism $\varphi$ is the whole space?

Thank you in advance for your reply.