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Math Help - Find all elements a+b(-3)^0.5 in Z[-3^0.5] such that |a+b(-3)^0.5|^2=1

  1. #1
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    Find all elements a+b(-3)^0.5 in Z[-3^0.5] such that |a+b(-3)^0.5|^2=1

    Hi,

    I have the following problem:

    "Find all elements a+b(-3)^{0.5} in Z[-3^{0.5}] such that |a+b(-3)^{0.5}|^2=1"

    And I think I have the answer, but I don't know if it is right since it says "elements" and not "element."

    Well, I know that |a+b(-3)^{0.5}|^2=a+b(-3)^{0.5} (obvious), and that if a+b(-3)^{0.5} is a unit, then |a+b(-3)^{0.5}|^2=1... is that correct?

    Thanks!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    (a+b\;\sqrt{-3})^2=a^2-3b^2+2ab\;\sqrt{-3}=1\Leftrightarrow a^2-3b^2=1\;\wedge\;2ab=0

    Now, analyze the cases a=0 and b=0 .
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  3. #3
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    Quote Originally Posted by mbmstudent View Post
    Hi,

    I have the following problem:

    "Find all elements a+b(-3)^{0.5} in Z[-3^{0.5}] such that |a+b(-3)^{0.5}|^2=1"

    And I think I have the answer, but I don't know if it is right since it says "elements" and not "element."

    Well, I know that |a+b(-3)^{0.5}|^2=a+b(-3)^{0.5} (obvious)


    Not ony not obvious but even wrong: |a+b\sqrt{-3}|^2=a^2+3b^2 , since we're talking here about

    the module (=absolute value) of a complex number




    , and that if a+b(-3)^{0.5} is a unit, then |a+b(-3)^{0.5}|^2=1... is that correct?

    Thanks!

    In fact, a^2+3b^2=1\Longleftrightarrow a=\pm 1 , so we have two different units here.

    Tonio
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  4. #4
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    Quote Originally Posted by FernandoRevilla View Post
    (a+b\;\sqrt{-3})^2=a^2-3b^2+2ab\;\sqrt{-3}=1\Leftrightarrow a^2-3b^2=1\;\wedge\;2ab=0

    Now, analyze the cases a=0 and b=0 .


    I think you missed the absolute value (module) symbol the OP wrote in his mesage.

    Tonio
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by tonio View Post
    I think you missed the absolute value (module) symbol the OP wrote in his mesage.

    Of course.
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