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Math Help - Algebraic Number Theory Help

  1. #1
    Susan
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    Algebraic Number Theory Help

    I am stuck on the following.

    Let R={(a+b(-163)^1/2)/2 : a,b are integers and a-b is even}. We know that R is a PID.

    Let z=(a+b(-163)^1/2)/2 then show that the cardinality of R/(z) is

    (a^2+b^2(163))/4

    this is just the magnitude of z squared but I am not sure how to arrive at the conclusion. Any help would be appreciated.

    Susan
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  2. #2
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    Quote Originally Posted by Susan View Post
    I am stuck on the following.

    Let R={(a+b(-163)^1/2)/2 : a,b are integers and a-b is even}. We know that R is a PID.

    Let z=(a+b(-163)^1/2)/2 then show that the cardinality of R/(z) is
    It makes no sense to let z=\frac{a}{2}+i\frac{b}{2}\sqrt{163}, how is that well defined? Note z has to be a specifc number in R because since it is a PID any ideal must have the form \left< z \right> where z is some particular element in R.
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