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Calculating the norm of an ideal in Z[√6]

  1. Mar 26, 2012 #1
    105s08x.png

    For part i) I deduced via Dedekind's criterion that

    <2> = <2,√6>2 & <3> = <3,√6>2

    So ii) I am trying to do now, and my arguement is thus:

    Let a be an ideal in Z[√6]. Suppose that N(a) = 24.

    By a proposition in my notes we have that

    a|<24> = <2,√6>6<3,√6>2

    so a = <2,√6>r<3,√6>s

    for some r in {0,1,2,3,4,5,6} and s in {0,1,2}

    We have N(a) = N(<2,√6>)rN(<3,√6>)s, and from that I can deduce what r and s should be.

    So now I need to calculate the norms of these two ideals.

    How do I calculate the norm of <2,√6>? Can I find a Z-basis for this ideal so that I can find a base change matrix and find the norm of that?

    Thanks
     
  2. jcsd
  3. Mar 28, 2012 #2

    morphism

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    In fact <2> and <3> are principal in Z[sqrt{6}].

    I'll explain why this is the case for <2>. First off, because <2>=<2,√6>^2, it suffices to show that <2,√6> is principal. For this, it suffices to exhibit an element of norm +/-2 (why?). So all we need to do is find a,b such that a^2-6b^2=+/-2, but this is pretty easy.
     
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