# Thread: Quick Question regarding Q Fields

1. ## Quick Question regarding Q Fields

Hello All.

I have been working on this problem and I"m closing in on the end. Here is where I'm at so far:

I've solved for $\displaystyle \lambda$ and I know that $\displaystyle \lambda = (3+\sqrt{-3})/2$ $\displaystyle \in$ $\displaystyle Q[\sqrt{-3}]$. I also know that $\displaystyle \lambda$ is a prime in $\displaystyle Q[\sqrt{-3}]$.

From here, I need some help with proving this:

If $\displaystyle \lambda$ divides $\displaystyle a$ for some rational integer $\displaystyle a$ in $\displaystyle Z$, it can be proven that 3 divides $\displaystyle a$.

Can someone show me how this is done using the rest of this info?

2. Can anyone apply my work to proving what I had asked? As far as the $\displaystyle \lambda$, I hope its clear that its $\displaystyle (3+\sqrt{-3})/2)$ and that it exists within $\displaystyle Q[\sqrt{-3}]$.

3. Originally Posted by Samson
Hello All.

I have been working on this problem and I"m closing in on the end. Here is where I'm at so far:

I've solved for $\displaystyle \lambda$ and I know that $\displaystyle \lambda = (3+\sqrt{-3})/2$ $\displaystyle \in$ $\displaystyle Q[\sqrt{-3}]$. I also know that $\displaystyle \lambda$ is a prime in $\displaystyle Q[\sqrt{-3}]$.

From here, I need some help with proving this:

If $\displaystyle \lambda$ divides $\displaystyle a$ for some rational integer $\displaystyle a$ in $\displaystyle Z$, it can be proven that 3 divides $\displaystyle a$.

Can someone show me how this is done using the rest of this info?
I think there must be something wrong with the problem. We are given lambda is prime, yet we know lambda divides lambda, therefore if the statement we want to prove is true, 3 divides lambda, which contradicts that lambda is prime.

4. I'm not exactly sure what you're saying. The way I'm looking at this is $\displaystyle \lambda | a$ and $\displaystyle 3 | a$. Doesn't this just mean that both $\displaystyle \lambda$ and $\displaystyle 3 |a$. Perhaps this might mean that $\displaystyle \lambda$ and $\displaystyle a$ are relatively prime?

5. Originally Posted by Samson
I'm not exactly sure what you're saying. The way I'm looking at this is $\displaystyle \lambda | a$ and $\displaystyle 3 | a$. Doesn't this just mean that both $\displaystyle \lambda$ and $\displaystyle 3 |a$. Perhaps this might mean that $\displaystyle \lambda$ and $\displaystyle a$ are relatively prime?
Sorry, I made a similar embarrassing mistake just a day or two ago, can't seem to learn my lesson..

I don't really have time to think about it right now, but you can just disregard my previous post.

6. Originally Posted by undefined
Sorry, I made a similar embarrassing mistake just a day or two ago, can't seem to learn my lesson..

I don't really have time to think about it right now, but you can just disregard my previous post.
Okay, well if you get the chance, I'd appreciate any help you can offer! I do appreciate your efforts thus far.