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One usually defines field extension $E/F$ whenever $F\subseteq E$. However, few authors would define field extension $F/K$ whenever there is a nonzero field homomorphism $F\rightarrow E$ (see e.g Definition 3.1 page 35 here:http://homepages.warwick.ac.uk/~masda/MA3D5/Galois.pdf.). Is there any (dis)advantage in defining like in the latter case? Will something be harder ( or missed) in doing so?

user136993
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    You’ll have to keep track of all the morphisms and all the different fields. Other than that, it’s a generalization and has all the classical advantages and disadvantages of generalizations: Some definitions, concepts and proofs become clearer, and with others you have to be more careful. For example, naturally a intermediate extension of $E/F$ would be an extension $L/F$ such that $E/L$ is an extension, too. Now, discerning these intermediate extensions becomes more subtle. And how can you talk about the “number” of intermediate extensions? One has to be more careful. – k.stm Apr 09 '14 at 20:53
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    Also, a field homomorphism $F\rightarrow E$ is also a ring homomorphism, so that the kernel of the homomorphism is an ideal of $F$. But since $F$ is a field, its only ideals are the zero ideal or the unit ideal. Since it's not zero, we know it isn't the unit ideal, and thus it is injective. Thus, $F$ can be regarded as a subfield of $E$. – Hayden Apr 09 '14 at 21:01
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    Maybe this link is relevant: http://math.stackexchange.com/questions/411755/this-tower-of-fields-is-being-ridiculous –  Apr 10 '14 at 04:16
  • @Hurkyl. Your link/answer brought me exactly the discussion I've been looking for. It is a pity that textbooks don't usually make any comment on this regard. – user136993 Apr 10 '14 at 13:10
  • @Hurkyl. Thanks a lot! Btw, How do select your answer as the official one? I'm new to this community. – user136993 Apr 10 '14 at 13:12
  • I hadn't made it an answer; I thought it only worth a comment. But if that really answered your question, I'll make it into an answer. –  Apr 11 '14 at 03:45

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This question demonstrates some of the issues that can come up when you're in the habit of reasoning about field extensions based on subsets, but you are dealing with a situation where the fields are in a different relationship than set-theoretic inclusion.

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