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Math Help - Golden Ratios (Q Fields)

  1. #1
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    Golden Ratios (Q Fields)

    Hello Everybody,

    As a continuation to my last thread, there are other examples that I struggled with.

    1. There exists a defining equation for the golden ratio: (1+Sqrt(5))/2 , and its norm in Q[Sqrt(5)] can also be found.

    Can anyone help me find this equation and the norm? I tried to think of it like normalizing vectors and using conjugates, but I don't think that its getting me anywhere.

    2. Assuming a and b are elements of Q[Sqrt(d)],, it can be shown that N(ab)=N(a)N(b) and N(a/b)=N(a)/N(b).

    How can this be shown? Should a exist in Q, how can we show that N(a)=(a^2) ?

    Any and all help is appreciated!
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  2. #2
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Samson View Post
    Hello Everybody,

    As a continuation to my last thread, there are other examples that I struggled with.

    1. There exists a defining equation for the golden ratio: (1+Sqrt(5))/2 , and its norm in Q[Sqrt(5)] can also be found.

    Can anyone help me find this equation and the norm? I tried to think of it like normalizing vectors and using conjugates, but I don't think that its getting me anywhere.

    2. Assuming a and b are elements of Q[Sqrt(d)],, it can be shown that N(ab)=N(a)N(b) and N(a/b)=N(a)/N(b).

    How can this be shown? Should a exist in Q, how can we show that N(a)=(a^2) ?

    Any and all help is appreciated!
    (1a) It is well known that the golden ratio is one of the roots of the polynomial x^2 - x - 1. It's on Wikipedia and many many websites. You can also do

    \displaystyle \left(x-\frac{1+\sqrt{5}}{2}\right)\left(x-\frac{1-\sqrt{5}}{2}\right)

    (1b) and (2) My Wikipedia-education is a bit slim on norms... It seems though that for \alpha=a+b\sqrt{D}, we have norm N(\alpha)=\alpha\overline{\alpha}. From this the other properties can probably be shown easily..?
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    Quote Originally Posted by undefined View Post
    (1a) It is well known that the golden ratio is one of the roots of the polynomial x^2 - x - 1. It's on Wikipedia and many many websites. You can also do

    \displaystyle \left(x-\frac{1+\sqrt{5}}{2}\right)\left(x-\frac{1-\sqrt{5}}{2}\right)

    (1b) and (2) My Wikipedia-education is a bit slim on norms... It seems though that for \alpha=a+b\sqrt{D}, we have norm N(\alpha)=\alpha\overline{\alpha}. From this the other properties can probably be shown easily..?
    Thank you undefined, I appreciate all the help! Is anybody else familiar enough with Norms to be able to pick this up at this point? Thank you!
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    Hey all, I just wanted to bump and see if anyone could explain those other properties like undefined had mentioned!

    All help is appreciated!
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  5. #5
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Samson View Post
    Hey all, I just wanted to bump and see if anyone could explain those other properties like undefined had mentioned!

    All help is appreciated!
    Read rule #1: http://www.mathhelpforum.com/math-he...carefully.html

    Odds are most people have seen this thread already.
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  6. #6
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    Quote Originally Posted by chiph588@ View Post
    Read rule #1: http://www.mathhelpforum.com/math-he...carefully.html

    Odds are most people have seen this thread already.
    Sorry! Does anybody know how we can do number two? I have tried following a few links on google as far as Norms go. Any help is appreciated!
    Here is one of the links I've found:

    http://www.jstor.org/pss/2324118
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  7. #7
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Samson View Post
    Sorry! Does anybody know how we can do number two? I have tried following a few links on google as far as Norms go. Any help is appreciated!
    Here is one of the links I've found:

    JSTOR: An Error Occurred Setting Your User Cookie
    Try a direct proof.
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  8. #8
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    Quote Originally Posted by chiph588@ View Post
    Try a direct proof.
    What do you mean by a "direct proof?"
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  9. #9
    MHF Contributor undefined's Avatar
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    Regarding forum rules: they're here to make everyone's lives easier really; consider post #4, which is the one that went against the rules:

    Quote Originally Posted by Samson View Post
    Hey all, I just wanted to bump and see if anyone could explain those other properties like undefined had mentioned!

    All help is appreciated!
    This adds nothing to the discussion because it gives us no useful information. All it does is tell us you think your question deserves more attention. From a helper standpoint, a post like this makes us less inclined to help you because then it's like we're working for you rather than with you. (Like we're slaves or something.) In addition it clutters the forum. So it's to everyone's disadvantage.

    Contrast this to post #6:

    Quote Originally Posted by Samson View Post
    Sorry! Does anybody know how we can do number two? I have tried following a few links on google as far as Norms go. Any help is appreciated!
    Here is one of the links I've found:

    JSTOR: An Error Occurred Setting Your User Cookie
    Here you give some more information and show effort in moving the discussion forward. This is within the rules. So if you feel the need to "bump" you should consider why you haven't gotten a sufficient response so far, and try to help us help you accordingly (for example by elaborating on any details that might not have been clear, giving us some definitions or references, responding to what we suggested, posting some work and saying where you got stuck, etc.).

    Now you haven't responded to my suggestion that N(\alpha)=\alpha\overline{\alpha}. I very much suspect it's right, but I was trying to get a definitive source and was piecing together info from here, here, here, here, here, and maybe others (I don't have number theory books easily available at the moment). Presumably there is a definition in your source that would allow you to easily verify that N(\alpha)=\alpha\overline{\alpha}.

    Now then, N(\alpha)=\alpha\overline{\alpha} = (a+b\sqrt{D})(a-b\sqrt{D}) = a^2 - b^2D. So

    N\left(\dfrac{1+\sqrt{5}}{2}\right) = \left(\dfrac{1}{2}\right)^2-\left(\dfrac{1}{2}\right)^2\cdot5 = -1

    And for your #2 questions, it's just algebra

    Let \alpha = a+b\sqrt{D}, \beta = c+d\sqrt{D}.

    Then \alpha\beta = (ac + bdD) + (ad + bc)\sqrt{D}

    N(\alpha\beta) = (ac + bdD)^2 - (ad + bc)^2D

    = a^2c^2+2abcdD+b^2d^2D^2-D(a^2d^2+2abcd+b^2c^2)

     = a^2c^2+b^2d^2D^2-a^2d^2D-b^2c^2D

    Compare with

    N(\alpha)N(\beta) = (a^2-b^2D)(c^2-d^2D)

    = a^2c^2-a^2d^2D-b^2c^2D+b^2d^2D^2

    See that they are the same.

    Do similarly for N(a/b)=N(a)/N(b).

    Quote Originally Posted by Samson View Post
    How can this be shown? Should a exist in Q, how can we show that N(a)=(a^2) ?
    We have \alpha = a + 0\sqrt{D}, so this question is trivial.
    Last edited by undefined; July 13th 2010 at 11:36 AM.
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  10. #10
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Samson View Post
    What do you mean by a "direct proof?"
    Direct proof - Wikipedia, the free encyclopedia
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    Thank you both very much! I really appreciate it! :-)
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