I need to show that, given [itex]F \subset E \subset K \subset L[/itex] (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 [itex]\phi:E \rightarrow L[/itex] that's the identity on F, that [itex]\phi(E) \subset K[/itex].
Edit: Yeah, if it wasn't obvious from the context, those are all supposed to be fields.
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 [itex]\alpha \in E-F[/itex] with [itex]\phi(\alpha) \in L-K[/itex], the additive and multiplicative inverses of both [itex]\alpha[/itex] and [itex]\phi(\alpha)[/itex] 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.