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Math Help - Quick Question regarding Q Fields

  1. #1
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    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 \lambda and I know that \lambda = (3+\sqrt{-3})/2 \in Q[\sqrt{-3}]. I also know that \lambda is a prime in Q[\sqrt{-3}].

    From here, I need some help with proving this:

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

    Can someone show me how this is done using the rest of this info?
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  2. #2
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    Can anyone apply my work to proving what I had asked? As far as the  \lambda , I hope its clear that its  (3+\sqrt{-3})/2) and that it exists within Q[\sqrt{-3}].
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  3. #3
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    Quote Originally Posted by Samson View Post
    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 \lambda and I know that \lambda = (3+\sqrt{-3})/2 \in Q[\sqrt{-3}]. I also know that \lambda is a prime in Q[\sqrt{-3}].

    From here, I need some help with proving this:

    If \lambda divides a for some rational integer a in Z, it can be proven that 3 divides 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.
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  4. #4
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    I'm not exactly sure what you're saying. The way I'm looking at this is \lambda | a and 3 | a. Doesn't this just mean that both  \lambda and 3 |a. Perhaps this might mean that \lambda and a are relatively prime?
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  5. #5
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    Quote Originally Posted by Samson View Post
    I'm not exactly sure what you're saying. The way I'm looking at this is \lambda | a and 3 | a. Doesn't this just mean that both  \lambda and 3 |a. Perhaps this might mean that \lambda and 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.
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  6. #6
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    Quote Originally Posted by undefined View Post
    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.

    Does anyone else have any suggestions on how to go about this?
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