Being dense about an Algebra problem

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SUMMARY

The discussion centers on proving that for fields F, E, K, and L, with K/F being a Galois extension and a homomorphism φ: E → L that is the identity on F, it follows that φ(E) ⊆ K. The user identifies that φ is injective due to its nature as a field homomorphism and contemplates the implications of elements α in E-F with φ(α) in L-K. The user expresses uncertainty about whether the Galois condition is necessary for the proof, indicating a need for clarity on the relationship between these fields and the homomorphism.

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Homework Statement



I need to show that, given F \subset E \subset K \subset L (K/F is Galois but I don't know how important that is for the part of the problem I'm having trouble with) and a homomorphism \phi:E \rightarrow L that's the identity on F, that \phi(E) \subset K.

Edit: Yeah, if it wasn't obvious from the context, those are all supposed to be fields.

Homework Equations

The Attempt at a Solution



So basically all the facts about this I've been able to come up with are that, being a field homomorphism, the map is injective, and that if I have \alpha \in E-F with \phi(\alpha) \in L-K, the additive and multiplicative inverses of both \alpha and \phi(\alpha) have to be in E-F and L-K respectively. I have a feeling I'm overlooking something really simple, but I just can't get my brain out of the funk to figure out what it is.
 
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I guess I'll give this a quick bump before I go to bed. Maybe I was wrong about missing something simple and I actually do have to use the Galois condition.
 
Yeah, I hate to keep doing this, but I still haven't figured it out. So I guess I'll try one more time. Sorry to be irritating.
 

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