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Math Help - Find units of Z[w]

  1. #1
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    Find units of Z[w]

    Let <br />
w = \frac {-1}{2} + \frac {3^{1/2}}{2}i<br />
, find units of Z[w].

    1 is the unity, so it is a unit. And what else...?
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let <br />
w = \frac {-1}{2} + \frac {3^{1/2}}{2}i<br />
, find units of Z[w].

    1 is the unity, so it is a unit. And what else...?
    The the previous excercises a+bw is a unit if and only if N(a+bw) = a^2+b^2 - ab = 1. Thus, a^2 - ab + (b^2 - 1) = 0 we require that the discrimant, b^2 - 4(b^2 - 4) > 0, thus, 3b^2 < 4 \implies b=0,1,-1, now you can solve for a.
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  3. #3
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    I don't understand how do you get to <br />
b^2 - 4(b^2 - 4) > 0<br />
, is there a general rule for that or you did by algebra? Further, I worked this out and I have 3b^2 < 16, not 4, did I do something wrong?

    Thanks.
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  4. #4
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    Quote Originally Posted by tttcomrader View Post
    I don't understand how do you get to <br />
b^2 - 4(b^2 - 4) > 0<br />
, is there a general rule for that or you did by algebra? Further, I worked this out and I have 3b^2 < 16, not 4, did I do something wrong?

    Thanks.
    Yes. I made a mistake it should be 3b^2 < 16. It is just the discrimant of the quadradic.
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