Being dense about an Algebra problem

[Mystic998]

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.

 

[Mystic998]

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.

 

[Mystic998]

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.