# Being dense about an Algebra problem

• Mystic998
In summary, the problem is to show that for a given chain of fields F \subset E \subset K \subset L where K/F is Galois, and a homomorphism \phi:E \rightarrow L that is the identity on F, we need to prove that \phi(E) \subset K. The steps taken so far include recognizing that \phi is injective and considering the elements \alpha \in E-F and \phi(\alpha) \in L-K, which require the knowledge of their respective additive and multiplicative inverses. The possibility of using the Galois condition is also mentioned. However, the solution has not been found yet and further attempts are needed.
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.

## 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.

Last edited:
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.

## 1. What does it mean to be "dense" about an Algebra problem?

Being dense about an Algebra problem means that you are struggling to understand or solve the problem. It can also refer to being stubborn or resistant to learning new methods or techniques.

## 2. How can I overcome being dense about an Algebra problem?

The best way to overcome being dense about an Algebra problem is to practice and seek help when needed. Don't be afraid to ask your teacher or a classmate for clarification or guidance.

## 3. Why do some people struggle with Algebra more than others?

Everyone has different strengths and weaknesses, and some people may find Algebra more challenging than others. It could also be due to a lack of understanding of fundamental concepts or not enough practice.

## 4. Is it normal to be dense about an Algebra problem?

Yes, it is normal to struggle with Algebra problems. It is a complex subject that requires logical thinking and problem-solving skills. With persistence and practice, anyone can improve their understanding of Algebra.

## 5. How can I prevent myself from being dense about an Algebra problem in the future?

To prevent being dense about an Algebra problem in the future, it is essential to review and reinforce your understanding of fundamental concepts regularly. It can also be helpful to break down difficult problems into smaller, more manageable steps and practice regularly.

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